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{{task|Arithmetic operations}} [[Category:Recursion]] [[Category:Memoization]] [[Category:Classic CS problems and programs]]

The '''Fibonacci sequence''' is a sequence Fn of natural numbers defined recursively:

  <big><big> F<sub>0</sub> = 0 </big></big>
<big><big> F<sub>1</sub> = 1 </big></big>
<big><big> F<sub>n</sub> = F<sub>n-1</sub> + F<sub>n-2</sub>, if n>1 </big></big>


;Task: Write a function to generate the nth Fibonacci number.

Solutions can be iterative or recursive (though recursive solutions are generally considered too slow and are mostly used as an exercise in recursion).

The sequence is sometimes extended into negative numbers by using a straightforward inverse of the positive definition:

  <big><big> F<sub>n</sub> = F<sub>n+2</sub> - F<sub>n+1</sub>, if n<0   </big></big>


support for negative n in the solution is optional.

• [[Fibonacci n-step number sequences‎]]
• [[Leonardo numbers]]

;References:

• [[wp:Fibonacci number|Wikipedia, Fibonacci number]]
• [[wp:Lucas number|Wikipedia, Lucas number]]
• [http://mathworld.wolfram.com/FibonacciNumber.html MathWorld, Fibonacci Number]
• [http://www.math-cs.ucmo.edu/~curtisc/articles/howardcooper/genfib4.pdf Some identities for r-Fibonacci numbers]
• [[oeis:A000045|OEIS Fibonacci numbers]]
• [[oeis:A000032|OEIS Lucas numbers]]

## 0815


%<:0D:>~$<:01:~%>=<:a94fad42221f2702:>~> }:_s:{x{={~$x+%{=>~>x~-x<:0D:~>~>~^:_s:?



## 360 Assembly

For maximum compatibility, programs use only the basic instruction set.

### using fullword integers

*        Fibonacci sequence    05/11/2014
*        integer (31 bits) = 10 decimals -> max fibo(46)
FIBONACC CSECT
USING FIBONACC,R12    base register
SAVEAREA B     STM-SAVEAREA(R15) skip savearea
DC    17F'0'          savearea
DC    CL8'FIBONACC'   eyecatcher
STM      STM   R14,R12,12(R13) save previous context
*        ----
LA    R1,0            f(n-2)=0
LA    R2,1            f(n-1)=1
LA    R4,2            n=2
LA    R6,1            step
LH    R7,NN           limit
LOOP     EQU   *               for n=2 to nn
LR    R3,R2             f(n)=f(n-1)
AR    R3,R1             f(n)=f(n-1)+f(n-2)
CVD   R4,PW             n  convert binary to packed (PL8)
UNPK  ZW,PW             packed (PL8) to zoned (ZL16)
MVC   CW,ZW             zoned (ZL16) to  char (CL16)
OI    CW+L'CW-1,X'F0'   zap sign
MVC   WTOBUF+5(2),CW+14 output
CVD   R3,PW             f(n) binary to packed decimal (PL8)
ED    ZN,PW             packed dec (PL8) to char (CL20)
MVC   WTOBUF+9(14),ZN+6 output
WTO   MF=(E,WTOMSG)     write buffer
LR    R1,R2             f(n-2)=f(n-1)
LR    R2,R3             f(n-1)=f(n)
BXLE  R4,R6,LOOP      endfor n
*        ----
LM    R14,R12,12(R13) restore previous savearea pointer
XR    R15,R15         return code set to 0
*        ----  DATA
NN       DC    H'46'           nn max n
PW       DS    PL8             15num
ZW       DS    ZL16
CW       DS    CL16
ZN       DS    CL20
*                  ' b 0 0 0 , 0 0 0 , 0 0 0 , 0 0 0 , 0 0 0'  15num
WTOMSG   DS    0F
DC    H'80',XL2'0000'
*                   fibo(46)=1836311903
WTOBUF   DC    CL80'fibo(12)=1234567890'
REGEQU
END   FIBONACC


{{out}}


...
fibo(41)=   165,580,141
fibo(42)=   267,914,296
fibo(43)=   433,494,437
fibo(44)=   701,408,733
fibo(45)= 1,134,903,170
fibo(46)= 1,836,311,903



### using packed decimals

*        Fibonacci sequence        31/07/2018
*        packed dec (PL8) = 15 decimals => max fibo(73)
FIBOWTOP CSECT
USING  FIBOWTOP,R13       base register
B      72(R15)            skip savearea
DC     17F'0'             savearea
SAVE   (14,12)            save previous context
*        ----
ZAP    FNM2,=P'0'         f(0)=0
ZAP    FNM1,=P'1'         f(1)=1
LA     R4,2               n=2
LA     R6,1               step
LH     R7,NN              limit
LOOP     EQU    *                  for n=2 to nn
ZAP    FN,FNM1              f(n)=f(n-2)
AP     FN,FNM2              f(n)=f(n-1)+f(n-2)
CVD    R4,PW                n
ED     ZN,PW                packed dec (PL8) to char (CL16)
MVC    WTOBUF+5(2),ZN+L'ZN-2  output
ED     ZN,FN                packed dec (PL8) to char (CL16)
MVC    WTOBUF+9(L'ZN),ZN        output
WTO    MF=(E,WTOMSG)        write buffer
ZAP    FNM2,FNM1            f(n-2)=f(n-1)
ZAP    FNM1,FN              f(n-1)=f(n)
BXLE   R4,R6,LOOP         endfor n
*        ----
L      R13,4(0,R13)       restore previous savearea pointer
RETURN (14,12),RC=0       restore registers from calling sav
*        ----   DATA
NN       DC     H'73'              nn
FNM2     DS     PL8                f(n-2)
FNM1     DS     PL8                f(n-1)
FN       DS     PL8                f(n)
PW       DS     PL8                15num
ZN       DS     CL20
*                   ' b 0 0 0 , 0 0 0 , 0 0 0 , 0 0 0 , 0 0 0'  15num
WTOMSG   DS     0F
DC     H'80',XL2'0000'
*                    fibo(73)=806515533049393
WTOBUF   DC     CL80'fibo(12)=123456789012345 '
REGEQU
END    FIBOWTOP


{{out}}


...
fibo(68)=  72,723,460,248,141
fibo(69)= 117,669,030,460,994
fibo(70)= 190,392,490,709,135
fibo(71)= 308,061,521,170,129
fibo(72)= 498,454,011,879,264
fibo(73)= 806,515,533,049,393



## 6502 Assembly

This subroutine stores the first n—by default the first ten—Fibonacci numbers in memory, beginning (because, why not?) at address 3867 decimal = F1B hex. Intermediate results are stored in three sequential addresses within the low 256 bytes of memory, which are the most economical to access.

The results are calculated and stored, but are not output to the screen or any other physical device: how to do that would depend on the hardware and the operating system.

       LDA  #0
STA  $F0 ; LOWER NUMBER LDA #1 STA$F1     ; HIGHER NUMBER
LDX  #0
LOOP:  LDA  $F1 STA$0F1B,X
STA  $F2 ; OLD HIGHER NUMBER ADC$F0
STA  $F1 ; NEW HIGHER NUMBER LDA$F2
STA  $F0 ; NEW LOWER NUMBER INX CPX #$0A    ; STOP AT FIB(10)
BMI  LOOP
RTS          ; RETURN FROM SUBROUTINE


## 8080 Assembly

This subroutine expects to be called with the value of $n$ in register A, and returns $f\left(n\right)$ also in A. You may want to take steps to save the previous contents of B, C, and D. The routine only works with fairly small values of $n$.

FIBNCI: MOV  C,  A  ; C will store the counter
DCR  C      ; decrement, because we know f(1) already
MVI  A,  1
MVI  B,  0
LOOP:   MOV  D,  A
ADD  B      ; A := A + B
MOV  B,  D
DCR  C
JNZ  LOOP   ; jump if not zero
RET         ; return from subroutine


## 8th

An iterative solution:


: fibon \ n -- fib(n)
>r 0 1
( tuck n:+ ) \ fib(n-2) fib(n-1) -- fib(n-1) fib(n)
r> n:1- times ;

: fib \ n -- fib(n)
dup 1 n:= if 1 ;; then
fibon nip ;



## ABAP

### Iterative

FORM fibonacci_iter USING index TYPE i
CHANGING number_fib TYPE i.
DATA: lv_old type i,
lv_cur type i.
Do index times.
If sy-index = 1 or sy-index = 2.
lv_cur = 1.
lv_old = 0.
endif.
number_fib = lv_cur + lv_old.
lv_old = lv_cur.
lv_cur = number_fib.
enddo.
ENDFORM.


### Impure Functional

{{works with|ABAP|7.4 SP08 Or above only}}

cl_demo_output=>display( REDUCE #( INIT fibnm = VALUE stringtab( ( |0| ) ( |1| ) )
n TYPE string
x = 0
y = 1
FOR i = 1 WHILE i <= 100
NEXT n = ( x + y )
fibnm = VALUE #( BASE fibnm ( n ) )
x = y
y = n ) ).


## ACL2

Fast, tail recursive solution:

(defun fast-fib-r (n a b)
(if (or (zp n) (zp (1- n)))
b
(fast-fib-r (1- n) b (+ a b))))

(defun fast-fib (n)
(fast-fib-r n 1 1))

(defun first-fibs-r (n i)
(declare (xargs :measure (nfix (- n i))))
(if (zp (- n i))
nil
(cons (fast-fib i)
(first-fibs-r n (1+ i)))))

(defun first-fibs (n)
(first-fibs-r n 0))


{{out}}

&gt;(first-fibs 20)
(1 1 2 3 5 8 13 21 34 55 89
144 233 377 610 987 1597 2584 4181 6765)



## ActionScript

public function fib(n:uint):uint
{
if (n < 2)
return n;

return fib(n - 1) + fib(n - 2);
}


### Recursive

with Ada.Text_IO, Ada.Command_Line;

procedure Fib is

function Fib(P: Positive) return Positive is
begin
if P <= 2 then
return 1;
else
return Fib(P-1) + Fib(P-2);
end if;
end Fib;

begin
Ada.Text_IO.Put("Fibonacci(" & Integer'Image(X) & " ) = ");
end Fib;


===Iterative, build-in integers===

with Ada.Text_IO;  use Ada.Text_IO;

procedure Test_Fibonacci is
function Fibonacci (N : Natural) return Natural is
This : Natural := 0;
That : Natural := 1;
Sum  : Natural;
begin
for I in 1..N loop
Sum  := This + That;
That := This;
This := Sum;
end loop;
return This;
end Fibonacci;
begin
for N in 0..10 loop
Put_Line (Positive'Image (Fibonacci (N)));
end loop;
end Test_Fibonacci;


{{out}}


0
1
1
2
3
5
8
13
21
34
55



===Iterative, long integers===

Using the big integer implementation from a cryptographic library [https://github.com/cforler/Ada-Crypto-Library/].

with Ada.Text_IO, Ada.Command_Line, Crypto.Types.Big_Numbers;

procedure Fibonacci is

Bit_Length: Positive := 1 + (696 * X) / 1000;
-- that number of bits is sufficient to store the full result.

package LN is new Crypto.Types.Big_Numbers
(Bit_Length + (32 - Bit_Length mod 32));
-- the actual number of bits has to be a multiple of 32
use LN;

function Fib(P: Positive) return Big_Unsigned is
Previous: Big_Unsigned := Big_Unsigned_Zero;
Result:   Big_Unsigned := Big_Unsigned_One;
Tmp:      Big_Unsigned;
begin
-- Result = 1 = Fibonacci(1)
for I in 1 .. P-1 loop
Tmp := Result;
Result := Previous + Result;
Previous := Tmp;
-- Result = Fibonacci(I+1))
end loop;
return Result;
end Fib;

begin
Ada.Text_IO.Put("Fibonacci(" & Integer'Image(X) & " ) = ");
end Fibonacci;


{{out}}

> ./fibonacci 777
Fibonacci( 777 ) = 1081213530912648191985419587942084110095342850438593857649766278346130479286685742885693301250359913460718567974798268702550329302771992851392180275594318434818082


### Fast method using fast matrix exponentiation



procedure fast_fibo is
-- We work with biggest natural integers in a 64 bits machine
type Big_Int is mod 2**64;

-- We provide an index type for accessing the fibonacci sequence terms
type Index is new Big_Int;

-- fibo is a generic function that needs a modulus type since it will return
-- the n'th term of the fibonacci sequence modulus this type (use Big_Int to get the
-- expected behaviour in this particular task)
generic
type ring_element is mod <>;
with function "*" (a, b : ring_element) return ring_element is <>;
function fibo (n : Index) return ring_element;
function fibo (n : Index) return ring_element is

type matrix is array (1 .. 2, 1 .. 2) of ring_element;

-- f is the matrix you apply to a column containing (F_n, F_{n+1}) to get
-- the next one containing (F_{n+1},F_{n+2})
-- could be a more general matrix (given as a generic parameter) to deal with
-- other linear sequences of order 2
f : constant matrix := (1 => (0, 1), 2 => (1, 1));

function "*" (a, b : matrix) return matrix is
(1 => (a(1,1)*b(1,1)+a(1,2)*b(2,1), a(1,1)*b(1,2)+a(1,2)*b(2,2)),
2 => (a(2,1)*b(1,1)+a(2,2)*b(2,1), a(2,1)*b(1,2)+a(2,2)*b(2,2)));

function square (m : matrix) return matrix is (m * m);

-- Fast_Pow could be non recursive but it doesn't really matter since
-- the number of calls is bounded up by the size (in bits) of Big_Int (e.g 64)
function fast_pow (m : matrix; n : Index) return matrix is
(if n = 0 then (1 => (1, 0), 2 => (0, 1)) -- = identity matrix
elsif n mod 2 = 0 then square (fast_pow (m, n / 2))
else m * square (fast_pow (m, n / 2)));

begin
return fast_pow (f, n)(2, 1);
end fibo;

function Big_Int_Fibo is new fibo (Big_Int);
begin
-- calculate instantly F_n with n=10^15 (modulus 2^64 )
put_line (Big_Int_Fibo (10**15)'img);
end fast_fibo;


### Recursive


#include "totvs.ch"
User Function fibb(a,b,n)
return(if(--n>0,fibb(b,a+b,n),a))



### Iterative


#include "totvs.ch"
User Function fibb(n)
local fnow:=0, fnext:=1, tempf
while (--n>0)
tempf:=fnow+fnext
fnow:=fnext
fnext:=tempf
end while
return(fnext)



## Aime

integer
fibs(integer n)
{
integer w;

if (n == 0) {
w = 0;
} elif (n == 1) {
w = 1;
} else {
integer a, b, i;

i = 1;
a = 0;
b = 1;
while (i < n) {
w = a + b;
a = b;
b = w;
i += 1;
}
}

return w;
}



## ALGOL 68

### Analytic

{{works with|ALGOL 68|Revision 1 - no extensions to language used}} {{works with|ALGOL 68G|Any - tested with release [http://sourceforge.net/projects/algol68/files/algol68g/algol68g-1.18.0/algol68g-1.18.0-9h.tiny.el5.centos.fc11.i386.rpm/download 1.18.0-9h.tiny]}} {{works with|ELLA ALGOL 68|Any (with appropriate job cards) - tested with release [http://sourceforge.net/projects/algol68/files/algol68toc/algol68toc-1.8.8d/algol68toc-1.8-8d.fc9.i386.rpm/download 1.8-8d]}}

PROC analytic fibonacci = (LONG INT n)LONG INT:(
LONG REAL sqrt 5 = long sqrt(5);
LONG REAL p = (1 + sqrt 5) / 2;
LONG REAL q = 1/p;
ROUND( (p**n + q**n) / sqrt 5 )
);

FOR i FROM 1 TO 30 WHILE
print(whole(analytic fibonacci(i),0));
# WHILE # i /= 30 DO
print(", ")
OD;
print(new line)


{{out}}


1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040



### Iterative

{{works with|ALGOL 68|Revision 1 - no extensions to language used}} {{works with|ALGOL 68G|Any - tested with release [http://sourceforge.net/projects/algol68/files/algol68g/algol68g-1.18.0/algol68g-1.18.0-9h.tiny.el5.centos.fc11.i386.rpm/download 1.18.0-9h.tiny]}} {{works with|ELLA ALGOL 68|Any (with appropriate job cards) - tested with release [http://sourceforge.net/projects/algol68/files/algol68toc/algol68toc-1.8.8d/algol68toc-1.8-8d.fc9.i386.rpm/download 1.8-8d]}}

PROC iterative fibonacci = (INT n)INT:
CASE n+1 IN
0, 1, 1, 2, 3, 5
OUT
INT even:=3, odd:=5;
FOR i FROM odd+1 TO n DO
(ODD i|odd|even) := odd + even
OD;
(ODD n|odd|even)
ESAC;

FOR i FROM 0 TO 30 WHILE
print(whole(iterative fibonacci(i),0));
# WHILE # i /= 30 DO
print(", ")
OD;
print(new line)


{{out}}


0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040



### Recursive

{{works with|ALGOL 68|Revision 1 - no extensions to language used}} {{works with|ALGOL 68G|Any - tested with release [http://sourceforge.net/projects/algol68/files/algol68g/algol68g-1.18.0/algol68g-1.18.0-9h.tiny.el5.centos.fc11.i386.rpm/download 1.18.0-9h.tiny]}} {{works with|ELLA ALGOL 68|Any (with appropriate job cards) - tested with release [http://sourceforge.net/projects/algol68/files/algol68toc/algol68toc-1.8.8d/algol68toc-1.8-8d.fc9.i386.rpm/download 1.8-8d]}}

PROC recursive fibonacci = (INT n)INT:
( n < 2 | n | fib(n-1) + fib(n-2));


### Generative

{{trans|Python|Note: This specimen retains the original [[Prime decomposition#Python|Python]] coding style.}} {{works with|ALGOL 68|Revision 1 - no extensions to language used}} {{works with|ALGOL 68G|Any - tested with release [http://sourceforge.net/projects/algol68/files/algol68g/algol68g-1.18.0/algol68g-1.18.0-9h.tiny.el5.centos.fc11.i386.rpm/download 1.18.0-9h.tiny]}} {{works with|ELLA ALGOL 68|Any (with appropriate job cards) - tested with release [http://sourceforge.net/projects/algol68/files/algol68toc/algol68toc-1.8.8d/algol68toc-1.8-8d.fc9.i386.rpm/download 1.8-8d]}}

MODE YIELDINT = PROC(INT)VOID;

PROC gen fibonacci = (INT n, YIELDINT yield)VOID: (
INT even:=0, odd:=1;
yield(even);
yield(odd);
FOR i FROM odd+1 TO n DO
yield( (ODD i|odd|even) := odd + even )
OD
);

main:(
# FOR INT n IN # gen fibonacci(30, # ) DO ( #
##   (INT n)VOID:(
print((" ",whole(n,0)))
# OD # ));
print(new line)
)


{{out}}


1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040



===Array (Table) Lookup=== {{works with|ALGOL 68|Revision 1 - no extensions to language used}} {{works with|ALGOL 68G|Any - tested with release [http://sourceforge.net/projects/algol68/files/algol68g/algol68g-1.18.0/algol68g-1.18.0-9h.tiny.el5.centos.fc11.i386.rpm/download 1.18.0-9h.tiny]}} {{works with|ELLA ALGOL 68|Any (with appropriate job cards) - tested with release [http://sourceforge.net/projects/algol68/files/algol68toc/algol68toc-1.8.8d/algol68toc-1.8-8d.fc9.i386.rpm/download 1.8-8d]}}

This uses a pre-generated list, requiring much less run-time processor usage, but assumes that INT is only 31 bits wide.

[]INT const fibonacci = []INT( -1836311903, 1134903170,
-701408733, 433494437, -267914296, 165580141, -102334155,
63245986, -39088169, 24157817, -14930352, 9227465, -5702887,
3524578, -2178309, 1346269, -832040, 514229, -317811, 196418,
-121393, 75025, -46368, 28657, -17711, 10946, -6765, 4181,
-2584, 1597, -987, 610, -377, 233, -144, 89, -55, 34, -21, 13,
-8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,
144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711,
28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040,
1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817,
39088169, 63245986, 102334155, 165580141, 267914296, 433494437,
701408733, 1134903170, 1836311903
)[@-46];

PROC VOID value error := stop;

PROC lookup fibonacci = (INT i)INT: (
IF LWB const fibonacci <= i AND i<= UPB const fibonacci THEN
const fibonacci[i]
ELSE
value error; SKIP
FI
);

FOR i FROM 0 TO 30 WHILE
print(whole(lookup fibonacci(i),0));
# WHILE # i /= 30 DO
print(", ")
OD;
print(new line)


{{out}}


0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040



=={{header|ALGOL-M}}== Note that the 21st Fibonacci number (= 10946) is the largest that can be calculated without overflowing ALGOL-M's integer data type.

### =Iterative=

INTEGER FUNCTION FIBONACCI( X ); INTEGER X;
BEGIN
INTEGER M, N, A, I;
M := 0;
N := 1;
FOR I := 2 STEP 1 UNTIL X DO
BEGIN
A := N;
N := M + N;
M := A;
END;
FIBONACCI := N;
END;


### =Naively recursive=

INTEGER FUNCTION FIBONACCI( X ); INTEGER X;
BEGIN
IF X < 3 THEN
FIBONACCI := 1
ELSE
FIBONACCI := FIBONACCI( X - 2 ) + FIBONACCI( X - 1 );
END;


## ALGOL W

begin
% return the nth Fibonacci number %
integer procedure Fibonacci( integer value n ) ;
begin
integer fn, fn1, fn2;
fn2 := 1;
fn1 := 0;
fn  := 0;
for i := 1 until n do begin
fn  := fn1 + fn2;
fn2 := fn1;
fn1 := fn
end ;
fn
end Fibonacci ;

for i := 0 until 10 do writeon( i_w := 3, s_w := 0, Fibonacci( i ) )

end.


{{out}}


0  1  1  2  3  5  8 13 21 34 55



## Alore

def fib(n as Int) as Int
if n < 2
return 1
end
return fib(n-1) + fib(n-2)
end


## AntLang

/Sequence
fib:{<0;1> {x,<x[-1]+x[-2]>}/ range[x]}
/nth
fibn:{fib[x][x]}


## Apex


/*
author: snugsfbay
date: March 3, 2016
description: Create a list of x numbers in the Fibonacci sequence.
- user may specify the length of the list
- enforces a minimum of 2 numbers in the sequence because any fewer is not a sequence
- enforces a maximum of 47 because further values are too large for integer data type
- Fibonacci sequence always starts with 0 and 1 by definition
*/
public class FibNumbers{

final static Integer MIN = 2; //minimum length of sequence
final static Integer MAX = 47; //maximum length of sequence

/*
description: method to create a list of numbers in the Fibonacci sequence
param: user specified integer representing length of sequence should be 2-47, inclusive.
- Sequence starts with 0 and 1 by definition so the minimum length could be as low as 2.
- For 48th number in sequence or greater, code would require a Long data type rather than an Integer.
return: list of integers in sequence.
*/
public static List<Integer> makeSeq(Integer len){

List<Integer> fib = new List<Integer>{0,1}; // initialize list with first two values
Integer i;

if(len<MIN || len==null || len>MAX) {
if (len>MAX){
len=MAX; //set length to maximum if user entered too high a value
}else{
len=MIN; //set length to minimum if user entered too low a value or none
}
} //This could be refactored using teneray operator, but we want code coverage to be reflected for each condition

//start with initial list size to find previous two values in the sequence, continue incrementing until list reaches user defined length
for(i=fib.size(); i<len; i++){
fib.add(fib[i-1]+fib[i-2]); //create new number based on previous numbers and add that to the list
}

return fib;
}

}



=

## Dyalog APL

=

### =Naive Recursive=


fib←{⍵≤1:⍵ ⋄ (∇ ⍵-1)+∇ ⍵-2}



Read this as: In the variable "fib", store the function that says, if the argument is less than or equal to 1, return the argument. Else, calculate the value you get when you recursively call the current function with the argument of the current argument minus one and add that to the value you get when you recursively call the current function with the argument of the current function minus two.

This naive solution requires Dyalog APL because GNU APL does not support this syntax for conditional guards.

=

## GNU APL/Dyalog APL

=

### =Array=

Since APL is an array language we'll use the following identity: :$\begin\left\{pmatrix\right\} 1 & 1 \ 1 & 0 \end\left\{pmatrix\right\}^n = \begin\left\{pmatrix\right\} F_\left\{n+1\right\} & F_n \ F_n & F_\left\{n-1\right\} \end\left\{pmatrix\right\}.$ In APL:


↑+.×/N/⊂2 2⍴1 1 1 0



Plugging in 4 for N gives the following result: :$\begin\left\{pmatrix\right\} 5 & 3 \ 3 & 2 \end\left\{pmatrix\right\}$ Here's what happens: We replicate the 2-by-2 matrix N times and then apply inner product-replication. The ''First'' removes the shell from the ''Enclose''. At this point we're basically done, but we need to pick out only $F_n$ in order to complete the task. Here's one way:


↑0 1↓↑+.×/N/⊂2 2⍴1 1 1 0



### =Analytic=

An alternative approach, using Binet's formula (which was apparently known long before Binet):

⌊.5+(((1+PHI)÷2)*⍳N)÷PHI←5*.5


## AppleScript

### Imperative

set fibs to {}
set x to (text returned of (display dialog "What fibbonaci number do you want?" default answer "3"))
set x to x as integer
repeat with y from 1 to x
if (y = 1 or y = 2) then
copy 1 to the end of fibs
else
copy ((item (y - 1) of fibs) + (item (y - 2) of fibs)) to the end of fibs
end if
end repeat
return item x of fibs


### Functional

The simple recursive version is famously slow:

on fib(n)
if n < 1 then
0
else if n < 3 then
1
else
fib(n - 2) + fib(n - 1)
end if
end fib


but we can combine '''enumFromTo(m, n)''' with the accumulator of a higher-order '''fold/reduce''' function to memoize the series:

{{Trans|JavaScript}} (ES6 memoized fold example) {{Trans|Haskell}} (Memoized fold example)

-- fib :: Int -> Int
on fib(n)

-- lastTwo : (Int, Int) -> (Int, Int)
script lastTwo
on |λ|([a, b])
[b, a + b]
end |λ|
end script

item 1 of foldl(lastTwo, {0, 1}, enumFromTo(1, n))
end fib

-- TEST -----------------------------------------------------------------------
on run

fib(32)

--> 2178309
end run

-- GENERIC FUNCTIONS ----------------------------------------------------------

-- enumFromTo :: Int -> Int -> [Int]
on enumFromTo(m, n)
if n < m then
set d to -1
else
set d to 1
end if
set lst to {}
repeat with i from m to n by d
set end of lst to i
end repeat
return lst
end enumFromTo

-- foldl :: (a -> b -> a) -> a -> [b] -> a
on foldl(f, startValue, xs)
tell mReturn(f)
set v to startValue
set lng to length of xs
repeat with i from 1 to lng
set v to |λ|(v, item i of xs, i, xs)
end repeat
return v
end tell
end foldl

-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: Handler -> Script
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn


{{Out}}

2178309


## Arendelle

( fibonacci , 1; 1 )

[ 98 , // 100 numbers of fibonacci

( fibonacci[ @fibonacci? ] ,

@fibonacci[ @fibonacci - 1 ] + @fibonacci[ @fibonacci - 2 ]

)

"Index: | @fibonacci? | => | @fibonacci[ @fibonacci? - 1 ] |"
]


## ARM Assembly

Expects to be called with $n$ in R0, and will return $f\left(n\right)$ in the same register.

fibonacci:
push  {r1-r3}
mov   r1,  #0
mov   r2,  #1

fibloop:
mov   r3,  r2
mov   r1,  r3
sub   r0,  r0,  #1
cmp   r0,  #1
bne   fibloop

mov   r0,  r2
pop   {r1-r3}
mov   pc,  lr


## ArnoldC

IT'S SHOWTIME

HEY CHRISTMAS TREE f1
YOU SET US UP @I LIED
TALK TO THE HAND f1

HEY CHRISTMAS TREE f2
YOU SET US UP @NO PROBLEMO

HEY CHRISTMAS TREE f3
YOU SET US UP @I LIED

STICK AROUND @NO PROBLEMO

GET TO THE CHOPPER f3
HERE IS MY INVITATION f1
GET UP f2
ENOUGH TALK
TALK TO THE HAND f3

GET TO THE CHOPPER f1
HERE IS MY INVITATION f2
ENOUGH TALK

GET TO THE CHOPPER f2
HERE IS MY INVITATION f3
ENOUGH TALK

CHILL

YOU HAVE BEEN TERMINATED


## Arturo

### Recursive

Fib [x]{
if x<2 { 1 }{
$(Fib x-1) +$(Fib x-2)
}
}


|      |
>-*>{+}/
| \+-/
1  |
#  1
|  #
|  |
.  .



## ATS

### Recursive


fun fib_rec(n: int): int =
if n >= 2 then fib_rec(n-1) + fib_rec(n-2) else n



### Iterative


(*
** This one is also referred to as being tail-recursive
*)
fun
fib_trec(n: int): int =
if
n > 0
then (fix loop (i:int, r0:int, r1:int): int => if i > 1 then loop (i-1, r1, r0+r1) else r1)(n, 0, 1)
else 0



### Iterative and Verified


(*
** This implementation is verified!
*)

dataprop FIB (int, int) =
| FIB0 (0, 0) | FIB1 (1, 1)
| {n:nat} {r0,r1:int} FIB2 (n+2, r0+r1) of (FIB (n, r0), FIB (n+1, r1))
// end of [FIB] // end of [dataprop]

fun
fibats{n:nat}
(n: int (n))
: [r:int] (FIB (n, r) | int r) = let
fun loop
{i:nat | i <= n}{r0,r1:int}
(
pf0: FIB (i, r0), pf1: FIB (i+1, r1)
| ni: int (n-i), r0: int r0, r1: int r1
) : [r:int] (FIB (n, r) | int r) =
if (ni > 0)
then loop{i+1}(pf1, FIB2 (pf0, pf1) | ni - 1, r1, r0 + r1)
else (pf0 | r0)
// end of [if]
// end of [loop]
in
loop {0} (FIB0 (), FIB1 () | n, 0, 1)
end // end of [fibats]



===Matrix-based===


(* ****** ****** *)
//
// How to compile:
// patscc -o fib fib.dats
//
(* ****** ****** *)
//
#include
//
(* ****** ****** *)
//
abst@ype
int3_t0ype =
(int, int, int)
//
typedef int3 = int3_t0ype
//
(* ****** ****** *)

extern
fun int3 : (int, int, int) -<> int3
extern
fun int3_1 : int3 -<> int
extern
fun mul_int3_int3: (int3, int3) -<> int3

(* ****** ****** *)

local

assume
int3_t0ype = (int, int, int)

in (* in-of-local *)
//
implement
int3 (x, y, z) = @(x, y, z)
//
implement int3_1 (xyz) = xyz.1
//
implement
mul_int3_int3
(
@(a,b,c), @(d,e,f)
) =
(a*d + b*e, a*e + b*f, b*e + c*f)
//
end // end of [local]

(* ****** ****** *)
//
implement
gnumber_int<int3> (n) = int3(n, 0, n)
//
implement gmul_val<int3> = mul_int3_int3
//
(* ****** ****** *)
//
fun
fib (n: intGte(0)): int =
int3_1(gpow_int_val<int3> (n, int3(1, 1, 0)))
//
(* ****** ****** *)

implement
main0 () =
{
//
val N = 10
val () = println! ("fib(", N, ") = ", fib(N))
val N = 20
val () = println! ("fib(", N, ") = ", fib(N))
val N = 30
val () = println! ("fib(", N, ") = ", fib(N))
val N = 40
val () = println! ("fib(", N, ") = ", fib(N))
//
} (* end of [main0] *)



## AutoHotkey

{{AutoHotkey case}}

### Iterative

{{trans|C}}

Loop, 5
MsgBox % fib(A_Index)
Return

fib(n)
{
If (n < 2)
Return n
i := last := this := 1
While (i <= n)
{
new := last + this
last := this
this := new
i++
}
Return this
}


### Recursive and iterative

Source: [http://www.autohotkey.com/forum/topic44657.html AutoHotkey forum] by Laszlo

/*
Important note: the recursive version would be very slow
without a global or static array. The iterative version
handles also negative arguments properly.
*/

FibR(n) {       ; n-th Fibonacci number (n>=0, recursive with static array Fibo)
Static
Return n<2 ? n : Fibo%n% ? Fibo%n% : Fibo%n% := FibR(n-1)+FibR(n-2)
}

Fib(n) {        ; n-th Fibonacci number (n < 0 OK, iterative)
a := 0, b := 1
Loop % abs(n)-1
c := b, b += a, a := c
Return n=0 ? 0 : n>0 || n&1 ? b : -b
}


## AutoIt

### Iterative

#AutoIt Version: 3.2.10.0
$n0 = 0$n1 = 1
$n = 10 MsgBox (0,"Iterative Fibonacci ", it_febo($n0,$n1,$n))

Func it_febo($n_0,$n_1,$N)$first = $n_0$second = $n_1$next = $first +$second
$febo = 0 For$i = 1 To $N-3$first = $second$second = $next$next = $first +$second
Next
if $n==0 Then$febo = 0
ElseIf $n==1 Then$febo = $n_0 ElseIf$n==2 Then
$febo =$n_1
Else
$febo =$next
EndIf
Return $febo EndFunc  ### Recursive #AutoIt Version: 3.2.10.0$n0 = 0
$n1 = 1$n = 10
MsgBox (0,"Recursive Fibonacci ", rec_febo($n0,$n1,$n)) Func rec_febo($r_0,$r_1,$R)
if  $R<3 Then if$R==2 Then
Return $r_1 ElseIf$R==1 Then
Return $r_0 ElseIf$R==0 Then
Return 0
EndIf
10
fib(10)=55


## Axe

A recursive solution is not practical in Axe because there is no concept of variable scope in Axe.

Iterative solution:

Lbl FIB
r₁→N
0→I
1→J
For(K,1,N)
I+J→T
J→I
T→J
End
J
Return


## bash

### Iterative


$fib=1;j=1;while((fib<100));do echo$fib;((k=fib+j,fib=j,j=k));done



1
1
2
3
5
8
13
21
34
55
89



### Recursive

fib()
{
if [ $1 -le 0 ] then echo 0 return 0 fi if [$1 -le 2 ]
then
echo 1
else
a=$(fib$[$1-1]) b=$(fib $[$1-2])

### Iterative

(fib=
last i this new
.   !arg:<2
|   0:?last:?i
& 1:?this
&   whl
' ( !i+1:<!arg:?i
& !last+!this:?new
& !this:?last
& !new:?this
)
& !this
)


fib$777 1081213530912648191985419587942084110095342850438593857649766278346130479286685742885693301250359913460718567974798268702550329302771992851392180275594318434818082 ## Brat ### Recursive fibonacci = { x | true? x < 2, x, { fibonacci(x - 1) + fibonacci(x - 2) } }  ### Tail Recursive fib_aux = { x, next, result | true? x == 0, result, { fib_aux x - 1, next + result, next } } fibonacci = { x | fib_aux x, 1, 0 }  ### Memoization cache = hash.new fibonacci = { x | true? cache.key?(x) { cache[x] } {true? x < 2, x, { cache[x] = fibonacci(x - 1) + fibonacci(x - 2) }} }  ## Burlesque  {0 1}{^^++[+[-^^-]\/}30.*\[e!vv   0 1{{.+}c!}{1000.<}w!  ## C ### Recursive long long fibb(long long a, long long b, int n) { return (--n>0)?(fibb(b, a+b, n)):(a); }  ### Iterative long long int fibb(int n) { int fnow = 0, fnext = 1, tempf; while(--n>0){ tempf = fnow + fnext; fnow = fnext; fnext = tempf; } return fnext; }  ### Analytic #include <tgmath.h> #define PHI ((1 + sqrt(5))/2) long long unsigned fib(unsigned n) { return floor( (pow(PHI, n) - pow(1 - PHI, n))/sqrt(5) ); }  ### Generative {{trans|Python}} {{works with|gcc|version 4.1.2 20080704 (Red Hat 4.1.2-44)}} #include <stdio.h> typedef enum{false=0, true=!0} bool; typedef void iterator; #include <setjmp.h> /* declare label otherwise it is not visible in sub-scope */ #define LABEL(label) jmp_buf label; if(setjmp(label))goto label; #define GOTO(label) longjmp(label, true) /* the following line is the only time I have ever required "auto" */ #define FOR(i, iterator) { auto bool lambda(i); yield_init = (void *)&lambda; iterator; bool lambda(i) #define DO { #define YIELD(x) if(!yield(x))return #define BREAK return false #define CONTINUE return true #define OD CONTINUE; } } static volatile void *yield_init; /* not thread safe */ #define YIELDS(type) bool (*yield)(type) = yield_init iterator fibonacci(int stop){ YIELDS(int); int f[] = {0, 1}; int i; for(i=0; i<stop; i++){ YIELD(f[i%2]); f[i%2]=f[0]+f[1]; } } main(){ printf("fibonacci: "); FOR(int i, fibonacci(16)) DO printf("%d, ",i); OD; printf("...\n"); }  {{out}}  fibonacci: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, ...  ### Fast method for a single large value #include <iostream> #include <stdio.h> #include <gmp.h> typedef struct node node; struct node { int n; mpz_t v; node *next; }; #define CSIZE 37 node *cache[CSIZE]; // very primitive linked hash table node * find_cache(int n) { int idx = n % CSIZE; node *p; for (p = cache[idx]; p && p->n != n; p = p->next); if (p) return p; p = malloc(sizeof(node)); p->next = cache[idx]; cache[idx] = p; if (n < 2) { p->n = n; mpz_init_set_ui(p->v, 1); } else { p->n = -1; // -1: value not computed yet mpz_init(p->v); } return p; } mpz_t tmp1, tmp2; mpz_t *fib(int n) { int x; node *p = find_cache(n); if (p->n < 0) { p->n = n; x = n / 2; mpz_mul(tmp1, *fib(x-1), *fib(n - x - 1)); mpz_mul(tmp2, *fib(x), *fib(n - x)); mpz_add(p->v, tmp1, tmp2); } return &p->v; } int main(int argc, char **argv) { int i, n; if (argc < 2) return 1; mpz_init(tmp1); mpz_init(tmp2); for (i = 1; i < argc; i++) { n = atoi(argv[i]); if (n < 0) { printf("bad input: %s\n", argv[i]); continue; } // about 75% of time is spent in printing gmp_printf("%Zd\n", *fib(n)); } return 0; }  {{out}}  % ./a.out 0 1 2 3 4 5 1 1 2 3 5 8 % ./a.out 10000000 | wc -c # count length of output, including the newline 1919488  ## C++ Using unsigned int, this version only works up to 48 before fib overflows. #include <iostream> int main() { unsigned int a = 1, b = 1; unsigned int target = 48; for(unsigned int n = 3; n <= target; ++n) { unsigned int fib = a + b; std::cout << "F("<< n << ") = " << fib << std::endl; a = b; b = fib; } return 0; }  {{libheader|GMP}} This version does not have an upper bound. #include <iostream> #include <gmpxx.h> int main() { mpz_class a = mpz_class(1), b = mpz_class(1); mpz_class target = mpz_class(100); for(mpz_class n = mpz_class(3); n <= target; ++n) { mpz_class fib = b + a; if ( fib < b ) { std::cout << "Overflow at " << n << std::endl; break; } std::cout << "F("<< n << ") = " << fib << std::endl; a = b; b = fib; } return 0; }  Version using transform: #include <algorithm> #include <vector> #include <functional> #include <iostream> unsigned int fibonacci(unsigned int n) { if (n == 0) return 0; std::vector<int> v(n+1); v[1] = 1; transform(v.begin(), v.end()-2, v.begin()+1, v.begin()+2, std::plus<int>()); // "v" now contains the Fibonacci sequence from 0 up return v[n]; }  Far-fetched version using adjacent_difference: #include <numeric> #include <vector> #include <functional> #include <iostream> unsigned int fibonacci(unsigned int n) { if (n == 0) return 0; std::vector<int> v(n, 1); adjacent_difference(v.begin(), v.end()-1, v.begin()+1, std::plus<int>()); // "array" now contains the Fibonacci sequence from 1 up return v[n-1]; }  Version which computes at compile time with metaprogramming: #include <iostream> template <int n> struct fibo { enum {value=fibo<n-1>::value+fibo<n-2>::value}; }; template <> struct fibo<0> { enum {value=0}; }; template <> struct fibo<1> { enum {value=1}; }; int main(int argc, char const *argv[]) { std::cout<<fibo<12>::value<<std::endl; std::cout<<fibo<46>::value<<std::endl; return 0; }  The following version is based on fast exponentiation: #include <iostream> inline void fibmul(int* f, int* g) { int tmp = f[0]*g[0] + f[1]*g[1]; f[1] = f[0]*g[1] + f[1]*(g[0] + g[1]); f[0] = tmp; } int fibonacci(int n) { int f[] = { 1, 0 }; int g[] = { 0, 1 }; while (n > 0) { if (n & 1) // n odd { fibmul(f, g); --n; } else { fibmul(g, g); n >>= 1; } } return f[1]; } int main() { for (int i = 0; i < 20; ++i) std::cout << fibonacci(i) << " "; std::cout << std::endl; }  {{out}} 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 ### Using Zeckendorf Numbers The nth fibonacci is represented as Zeckendorf 1 followed by n-1 zeroes. [[Zeckendorf number representation#Using a C++11 User Defined Literal|Here]] I define a class N which defines the operations increment ++() and comparison <=(other N) for Zeckendorf Numbers.  // Use Zeckendorf numbers to display Fibonacci sequence. // Nigel Galloway October 23rd., 2012 int main(void) { char NG[22] = {'1',0}; int x = -1; N G; for (int fibs = 1; fibs <= 20; fibs++) { for (;G <= N(NG); ++G) x++; NG[fibs] = '0'; NG[fibs+1] = 0; std::cout << x << " "; } std::cout << std::endl; return 0; }  {{out}}  1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946  ### Using Standard Template Library Possibly less "Far-fetched version".  // Use Standard Template Library to display Fibonacci sequence. // Nigel Galloway March 30th., 2013 #include <algorithm> #include <iostream> #include <iterator> int main() { int x = 1, y = 1; generate_n(std::ostream_iterator<int>(std::cout, " "), 21, [&]{int n=x; x=y; y+=n; return n;}); return 0; }  {{out}} 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 ## C# ### Recursive  public static ulong Fib(uint n) { return (n < 2)? n : Fib(n - 1) + Fib(n - 2); }  === Tail-Recursive ===  public static ulong Fib(uint n) { return Fib(0, 1, n); } private static ulong Fib(ulong a, ulong b, uint n) { return (n < 1)? a :(n == 1)? b : Fib(b, a + b, n - 1); }  ### Iterative  public static ulong Fib(uint x) { if (x == 0) return 0; ulong prev = 0; ulong next = 1; for (int i = 1; i < x; i++) { ulong sum = prev + next; prev = next; next = sum; } return next; }  === Eager-Generative ===  public static IEnumerable<long> Fibs(uint x) { IList<ulong> fibs = new List<ulong>(); ulong prev = -1; ulong next = 1; for (int i = 0; i < x; i++) { long sum = prev + next; prev = next; next = sum; fibs.Add(sum); } return fibs; }  === Lazy-Generative ===  public static IEnumerable<ulong> Fibs(uint x) { ulong prev = -1; ulong next = 1; for (uint i = 0; i < x; i++) { ulong sum = prev + next; prev = next; next = sum; yield return sum; } }  ### Analytic Only works to the 92th fibonacci number.  private static double Phi = ((1d + Math.Sqrt(5d))/2d); private static double D = 1d/Math.Sqrt(5d); ulong Fib(uint n) { if(n > 92) throw new ArgumentOutOfRangeException("n", n, "Needs to be smaller than 93."); return (ulong)((Phi^n) - (1d - Phi)^n))*D); }  ### Matrix Algorithm is based on :$\begin\left\{pmatrix\right\}1&1\1&0\end\left\{pmatrix\right\}^n = \begin\left\{pmatrix\right\}F\left(n+1\right)&F\left(n\right)\F\left(n\right)&F\left(n-1\right)\end\left\{pmatrix\right\}$. Needs System.Windows.Media.Matrix or similar Matrix class. Calculates in $O\left(n\right)$.  public static ulong Fib(uint n) { var M = new Matrix(1,0,0,1); var N = new Matrix(1,1,1,0); for (uint i = 1; i < n; i++) M *= N; return (ulong)M[0][0]; }  Needs System.Windows.Media.Matrix or similar Matrix class. Calculates in $O\left(\log\left\{n\right\}\right)$.  private static Matrix M; private static readonly Matrix N = new Matrix(1,1,1,0); public static ulong Fib(uint n) { M = new Matrix(1,0,0,1); MatrixPow(n-1); return (ulong)M[0][0]; } private static void MatrixPow(double n){ if (n > 1) { MatrixPow(n/2); M *= M; } if (n % 2 == 0) M *= N; }  === Array (Table) Lookup ===  private static int[] fibs = new int[]{ -1836311903, 1134903170, -701408733, 433494437, -267914296, 165580141, -102334155, 63245986, -39088169, 24157817, -14930352, 9227465, -5702887, 3524578, -2178309, 1346269, -832040, 514229, -317811, 196418, -121393, 75025, -46368, 28657, -17711, 10946, -6765, 4181, -2584, 1597, -987, 610, -377, 233, -144, 89, -55, 34, -21, 13, -8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155, 165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903}; public static int Fib(int n) { if(n < -46 || n > 46) throw new ArgumentOutOfRangeException("n", n, "Has to be between -46 and 47.") return fibs[n+46]; }  ### Arbitrary Precision {{libheader|System.Numerics}} This large step recurrence routine can calculate the two millionth Fibonacci number in under 1 / 5 second at tio.run. This routine can generate the fifty millionth Fibonacci number in under 30 seconds at tio.run. The unused conventional iterative method times out at two million on tio.run, you can only go to around 1,290,000 or so to keep the calculation time (plus string conversion time) under the 60 second timeout limit there. When using this large step recurrence method, it takes around 5 seconds to convert the two millionth Fibonacci number (417975 digits) into a string (so that one may count those digits). using System; using System.Collections.Generic; using System.Numerics; static class QuikFib { // A sparse array of values calculated along the way private static SortedList<int, BigInteger> sl = new SortedList<int, BigInteger>(); // Square a BigInteger public static BigInteger sqr(BigInteger n) { return n * n; } // Helper routine for Fsl(). It adds an entry to the sorted list when necessary public static void IfNec(int n) { if (!sl.ContainsKey(n)) sl.Add(n, Fsl(n)); } // This routine is semi-recursive, but doesn't need to evaluate every number up to n. // Algorithm from here: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibFormula.html#section3 public static BigInteger Fsl(int n) { if (n < 2) return n; int n2 = n >> 1, pm = n2 + ((n & 1) << 1) - 1; IfNec(n2); IfNec(pm); return n2 > pm ? (2 * sl[pm] + sl[n2]) * sl[n2] : sqr(sl[n2]) + sqr(sl[pm]); } // Conventional iteration method (not used here) public static BigInteger Fm(BigInteger n) { if (n < 2) return n; BigInteger cur = 0, pre = 1; for (int i = 0; i <= n - 1; i++) { BigInteger sum = cur + pre; pre = cur; cur = sum; } return cur; } public static void Main() { int num = 2_000_000; DateTime st = DateTime.Now; BigInteger v = Fsl(num); Console.WriteLine("{0:n3} ms to calculate the {1:n0}th Fibonacci number,", (DateTime.Now - st).TotalMilliseconds, num); st = DateTime.Now; string vs = v.ToString(); Console.WriteLine("{0:n3} seconds to convert to a string.", (DateTime.Now - st).TotalSeconds); Console.WriteLine("number of digits is {0}", vs.Length); if (vs.Length < 10000) { st = DateTime.Now; Console.WriteLine(vs); Console.WriteLine("{0:n3} ms to write it to the console.", (DateTime.Now - st).TotalMilliseconds); } else Console.WriteLine("partial: {0}...{1}", vs.Substring(1, 35), vs.Substring(vs.Length - 35)); } }  {{out}} 179.978 ms to calculate the 2,000,000th Fibonacci number, 4.728 seconds to convert to a string. number of digits is 417975 partial: 53129491750764154305166065450382516...91799493108960825129188777803453125  ## Cat define fib { dup 1 <= [] [dup 1 - fib swap 2 - fib +] if }  ## Chapel iter fib() { var a = 0, b = 1; while true { yield a; (a, b) = (b, b + a); } }  ## Chef Stir-Fried Fibonacci Sequence. An unobfuscated iterative implementation. It prints the first N + 1 Fibonacci numbers, where N is taken from standard input. Ingredients. 0 g last 1 g this 0 g new 0 g input Method. Take input from refrigerator. Put this into 4th mixing bowl. Loop the input. Clean the 3rd mixing bowl. Put last into 3rd mixing bowl. Add this into 3rd mixing bowl. Fold new into 3rd mixing bowl. Clean the 1st mixing bowl. Put this into 1st mixing bowl. Fold last into 1st mixing bowl. Clean the 2nd mixing bowl. Put new into 2nd mixing bowl. Fold this into 2nd mixing bowl. Put new into 4th mixing bowl. Endloop input until looped. Pour contents of the 4th mixing bowl into baking dish. Serves 1.  ## CMake Iteration uses a while() loop. Memoization uses global properties. set_property(GLOBAL PROPERTY fibonacci_0 0) set_property(GLOBAL PROPERTY fibonacci_1 1) set_property(GLOBAL PROPERTY fibonacci_next 2) # var = nth number in Fibonacci sequence. function(fibonacci var n) # If the sequence is too short, compute more Fibonacci numbers. get_property(next GLOBAL PROPERTY fibonacci_next) if(NOT next GREATER${n})
# a, b = last 2 Fibonacci numbers
math(EXPR i "${next} - 2") get_property(a GLOBAL PROPERTY fibonacci_${i})
math(EXPR i "${next} - 1") get_property(b GLOBAL PROPERTY fibonacci_${i})

while(NOT next GREATER ${n}) math(EXPR i "${a} + ${b}") # i = next Fibonacci number set_property(GLOBAL PROPERTY fibonacci_${next} ${i}) set(a${b})
set(b ${i}) math(EXPR next "${next} + 1")
endwhile()
set_property(GLOBAL PROPERTY fibonacci_next ${next}) endif() get_property(answer GLOBAL PROPERTY fibonacci_${n})
set(${var}${answer} PARENT_SCOPE)
endfunction(fibonacci)

# Test program: print 0th to 9th and 25th to 30th Fibonacci numbers.
set(s "")
foreach(i RANGE 0 9)
fibonacci(f ${i}) set(s "${s} ${f}") endforeach(i) set(s "${s} ... ")
foreach(i RANGE 25 30)
fibonacci(f ${i}) set(s "${s} ${f}") endforeach(i) message(${s})

 0 1 1 2 3 5 8 13 21 34 ... 75025 121393 196418 317811 514229 832040


## Clio

Clio is pure and functions are lazy and memoized by default

fn fib n:
if n < 2: n
else: (n - 1 -> fib) + (n - 2 -> fib)

[0:100] -> * fib -> * print


## Clojure

### Lazy Sequence

This is implemented idiomatically as an infinitely long, lazy sequence of all Fibonacci numbers:

(defn fibs []
(map first (iterate (fn [[a b]] [b (+ a b)]) [0 1])))


Thus to get the nth one:

(nth (fibs) 5)


So long as one does not hold onto the head of the sequence, this is unconstrained by length.

The one-line implementation may look confusing at first, but on pulling it apart it actually solves the problem more "directly" than a more explicit looping construct.

(defn fibs []
(map first ;; throw away the "metadata" (see below) to view just the fib numbers
(iterate ;; create an infinite sequence of [prev, curr] pairs
(fn [[a b]] ;; to produce the next pair, call this function on the current pair
[b (+ a b)]) ;; new prev is old curr, new curr is sum of both previous numbers
[0 1]))) ;; recursive base case: prev 0, curr 1


A more elegant solution is inspired by the Haskell implementation of an infinite list of Fibonacci numbers:

(def fib (lazy-cat [0 1] (map + fib (rest fib))))


Then, to see the first ten,

 (take 10 fib)
(0 1 1 2 3 5 8 13 21 34)


### Iterative

Here's a simple interative process (using a recursive function) that carries state along with it (as args) until it reaches a solution:

;; max is which fib number you'd like computed (0th, 1st, 2nd, etc.)
;; n is which fib number you're on for this call (0th, 1st, 2nd, etc.)
;; j is the nth fib number (ex. when n = 5, j = 5)
;; i is the nth - 1 fib number
(defn- fib-iter
[max n i j]
(if (= n max)
j
(recur max
(inc n)
j
(+ i j))))

(defn fib
[max]
(if (< max 2)
max
(fib-iter max 1 0N 1N)))


"defn-" means that the function is private (for use only inside this library). The "N" suffixes on integers tell Clojure to use arbitrary precision ints for those.

===Doubling Algorithm (Fast)=== Based upon the doubling algorithm which computes in O(log (n)) time as described here https://www.nayuki.io/page/fast-fibonacci-algorithms Implementation credit: https://stackoverflow.com/questions/27466311/how-to-implement-this-fast-doubling-fibonacci-algorithm-in-clojure/27466408#27466408


(defn fib [n]
(letfn [(fib* [n]
(if (zero? n)
[0 1]
(let [[a b] (fib* (quot n 2))
c (*' a (-' (*' 2 b) a))
d (+' (*' b b) (*' a a))]
(if (even? n)
[c d]
[d (+' c d)]))))]
(first (fib* n))))



### Recursive

A naive slow recursive solution:

(defn fib [n]
(case n
0 0
1 1
(+ (fib (- n 1))
(fib (- n 2)))))


This can be improved to an O(n) solution, like the iterative solution, by memoizing the function so that numbers that have been computed are cached. Like a lazy sequence, this also has the advantage that subsequent calls to the function use previously cached results rather than recalculating.

(def fib
(memoize
(fn [n]
(case n
0 0
1 1
(+ (fib (- n 1))
(fib (- n 2)))))))


### Using core.async

(ns fib.core)
(require '[clojure.core.async
:refer [<! >! >!! <!! timeout chan alt! go]])

(defn fib [c]
(loop [a 0 b 1]
(>!! c a)
(recur b (+ a b))))

(defn -main []
(let [c (chan)]
(go (fib c))
(dorun
(for [i (range 10)]
(println (<!! c))))))



## COBOL

### Iterative

Program-ID. Fibonacci-Sequence.
Data Division.
Working-Storage Section.
01  FIBONACCI-PROCESSING.
05  FIBONACCI-NUMBER  PIC 9(36)   VALUE 0.
05  FIB-ONE           PIC 9(36)   VALUE 0.
05  FIB-TWO           PIC 9(36)   VALUE 1.
01  DESIRED-COUNT       PIC 9(4).
01  FORMATTING.
05  INTERM-RESULT     PIC Z(35)9.
05  FORMATTED-RESULT  PIC X(36).
05  FORMATTED-SPACE   PIC x(35).
Procedure Division.
000-START-PROGRAM.
Display "What place of the Fibonacci Sequence would you like (<173)? " with no advancing.
Accept DESIRED-COUNT.
If DESIRED-COUNT is less than 1
Stop run.
If DESIRED-COUNT is less than 2
Move FIBONACCI-NUMBER to INTERM-RESULT
Move INTERM-RESULT to FORMATTED-RESULT
Unstring FORMATTED-RESULT delimited by all spaces into FORMATTED-SPACE,FORMATTED-RESULT
Display FORMATTED-RESULT
Stop run.
Subtract 1 from DESIRED-COUNT.
Move FIBONACCI-NUMBER to INTERM-RESULT.
Move INTERM-RESULT to FORMATTED-RESULT.
Unstring FORMATTED-RESULT delimited by all spaces into FORMATTED-SPACE,FORMATTED-RESULT.
Display FORMATTED-RESULT.
Perform 100-COMPUTE-FIBONACCI until DESIRED-COUNT = zero.
Stop run.
100-COMPUTE-FIBONACCI.
Compute FIBONACCI-NUMBER = FIB-ONE + FIB-TWO.
Move FIB-TWO to FIB-ONE.
Move FIBONACCI-NUMBER to FIB-TWO.
Subtract 1 from DESIRED-COUNT.
Move FIBONACCI-NUMBER to INTERM-RESULT.
Move INTERM-RESULT to FORMATTED-RESULT.
Unstring FORMATTED-RESULT delimited by all spaces into FORMATTED-SPACE,FORMATTED-RESULT.
Display FORMATTED-RESULT.


### Recursive

{{works with|GNU Cobol|2.0}}

SOURCE FREE
IDENTIFICATION DIVISION.
PROGRAM-ID. fibonacci-main.

DATA DIVISION.
WORKING-STORAGE SECTION.
01  num                                 PIC 9(6) COMP.
01  fib-num                             PIC 9(6) COMP.

PROCEDURE DIVISION.
ACCEPT num
CALL "fibonacci" USING CONTENT num RETURNING fib-num
DISPLAY fib-num
.
END PROGRAM fibonacci-main.

IDENTIFICATION DIVISION.
PROGRAM-ID. fibonacci RECURSIVE.

DATA DIVISION.
LOCAL-STORAGE SECTION.
01  1-before                            PIC 9(6) COMP.
01  2-before                            PIC 9(6) COMP.

01  num                                 PIC 9(6) COMP.

01  fib-num                             PIC 9(6) COMP BASED.

PROCEDURE DIVISION USING num RETURNING fib-num.
ALLOCATE fib-num
EVALUATE num
WHEN 0
MOVE 0 TO fib-num
WHEN 1
MOVE 1 TO fib-num
WHEN OTHER
SUBTRACT 1 FROM num
CALL "fibonacci" USING CONTENT num RETURNING 1-before
SUBTRACT 1 FROM num
CALL "fibonacci" USING CONTENT num RETURNING 2-before
ADD 1-before TO 2-before GIVING fib-num
END-EVALUATE
.
END PROGRAM fibonacci.


## CoffeeScript

### Analytic

fib_ana = (n) ->
sqrt = Math.sqrt
phi = ((1 + sqrt(5))/2)
Math.round((Math.pow(phi, n)/sqrt(5)))


### Iterative

fib_iter = (n) ->
return n if n < 2
[prev, curr] = [0, 1]
[prev, curr] = [curr, curr + prev] for i in [1..n]
curr


### Recursive

fib_rec = (n) ->
if n < 2 then n else fib_rec(n-1) + fib_rec(n-2)


## Comefrom0x10

Recursion is is not possible in Comefrom0x10.

### Iterative

stop = 6
a = 1
i = 1  # start
a      # print result

fib
comefrom if i is 1  # start
b = 1
comefrom fib        # start of loop
i = i + 1
next_b = a + b
a = b
b = next_b

comefrom fib if i > stop


## Common Lisp

Note that Common Lisp uses bignums, so this will never overflow.

### Iterative

(defun fibonacci-iterative (n &aux (f0 0) (f1 1))
(case n
(0 f0)
(1 f1)
(t (loop for n from 2 to n
for a = f0 then b and b = f1 then result
for result = (+ a b)
finally (return result)))))


Simpler one:

(defun fibonacci (n)
(let ((a 0) (b 1) (c n))
(loop for i from 2 to n do
(setq c (+ a b)
a b
b c))
c))


Not a function, just printing out the entire (for some definition of "entire") sequence with a for var =  loop:

(loop for x = 0 then y and y = 1 then (+ x y) do (print x))


### Recursive

(defun fibonacci-recursive (n)
(if (< n 2)
n
(+ (fibonacci-recursive (- n 2)) (fibonacci-recursive (- n 1)))))

(defun fibonacci-tail-recursive ( n &optional (a 0) (b 1))
(if (= n 0)
a
(fibonacci-tail-recursive (- n 1) b (+ a b))))


Tail recursive and squaring:

(defun fib (n &optional (a 1) (b 0) (p 0) (q 1))
(if (= n 1) (+ (* b p) (* a q))
(fib (ash n -1)
(if (evenp n) a (+ (* b q) (* a (+ p q))))
(if (evenp n) b (+ (* b p) (* a q)))
(+ (* p p) (* q q))
(+ (* q q) (* 2 p q))))) ;p is Fib(2^n-1), q is Fib(2^n).

(print (fib 100000))


### Alternate solution


;; Project : Fibonacci sequence

(defun fibonacci (nr)
(cond ((= nr 0) 1)
((= nr 1) 1)
(t (+ (fibonacci (- nr 1))
(fibonacci (- nr 2))))))
(format t "~a" "First 10 Fibonacci numbers")
(dotimes (n 10)
(if (< n 1) (terpri))
(if (< n 9) (format t "~a" " "))
(write(+ n 1)) (format t "~a" ": ")
(write (fibonacci n)) (terpri))



Output:


First 10 Fibonacci numbers
1: 1
2: 1
3: 2
4: 3
5: 5
6: 8
7: 13
8: 21
9: 34
10: 55



### Solution with methods and eql specializers


(defmethod fib (n)
(declare ((integer 0 *) n))
(+ (fib (- n 1))
(fib (- n 2))))

(defmethod fib ((n (eql 0))) 0)

(defmethod fib ((n (eql 1))) 1)



## Computer/zero Assembly

To find the $n$th Fibonacci number, set the initial value of count equal to $n$–2 and run the program. The machine will halt with the answer stored in the accumulator. Since Computer/zero's word length is only eight bits, the program will not work with values of $n$ greater than 13.

loop:   LDA  y      ; higher No.
STA  temp
STA  y
LDA  temp
STA  x

LDA  count
SUB  one
BRZ  done

STA  count
JMP  loop

done:   LDA  y
STP

one:         1
count:       8      ; n = 10
x:           1
y:           1
temp:        0


## D

Here are four versions of Fibonacci Number calculating functions. ''FibD'' has an argument limit of magnitude 84 due to floating point precision, the others have a limit of 92 due to overflow (long).The traditional recursive version is inefficient. It is optimized by supplying a static storage to store intermediate results. A Fibonacci Number generating function is added. All functions have support for negative arguments.

import std.stdio, std.conv, std.algorithm, std.math;

long sgn(alias unsignedFib)(int n) { // break sign manipulation apart
immutable uint m = (n >= 0) ? n : -n;
if (n < 0 && (n % 2 == 0))
return -unsignedFib(m);
else
return unsignedFib(m);
}

long fibD(uint m) { // Direct Calculation, correct for abs(m) <= 84
enum sqrt5r =  1.0L / sqrt(5.0L);         //  1 / sqrt(5)
enum golden = (1.0L + sqrt(5.0L)) / 2.0L; // (1 + sqrt(5)) / 2
return roundTo!long(pow(golden, m) * sqrt5r);
}

long fibI(in uint m) pure nothrow { // Iterative
long thisFib = 0;
long nextFib = 1;
foreach (i; 0 .. m) {
long tmp = nextFib;
nextFib += thisFib;
thisFib  = tmp;
}
return thisFib;
}

long fibR(uint m) { // Recursive
return (m < 2) ? m : fibR(m - 1) + fibR(m - 2);
}

long fibM(uint m) { // memoized Recursive
static long[] fib = [0, 1];
while (m >= fib.length )
fib ~= fibM(m - 2) + fibM(m - 1);
return fib[m];
}

alias sgn!fibD sfibD;
alias sgn!fibI sfibI;
alias sgn!fibR sfibR;
alias sgn!fibM sfibM;

auto fibG(in int m) { // generator(?)
immutable int sign = (m < 0) ? -1 : 1;
long yield;

return new class {
final int opApply(int delegate(ref int, ref long) dg) {
int idx = -sign; // prepare for pre-increment
foreach (f; this)
if (dg(idx += sign, f))
break;
return 0;
}

final int opApply(int delegate(ref long) dg) {
long f0, f1 = 1;
foreach (p; 0 .. m * sign + 1) {
if (sign == -1 && (p % 2 == 0))
yield = -f0;
else
yield = f0;
if (dg(yield)) break;
auto temp = f1;
f1 = f0 + f1;
f0 = temp;
}
return 0;
}
};
}

void main(in string[] args) {
int k = args.length > 1 ? to!int(args[1]) : 10;
writefln("Fib(%3d) = ", k);
writefln("D : %20d <- %20d + %20d",
sfibD(k), sfibD(k - 1), sfibD(k - 2));
writefln("I : %20d <- %20d + %20d",
sfibI(k), sfibI(k - 1), sfibI(k - 2));
if (abs(k) < 36 || args.length > 2)
// set a limit for recursive version
writefln("R : %20d <- %20d + %20d",
sfibR(k), sfibM(k - 1), sfibM(k - 2));
writefln("O : %20d <- %20d + %20d",
sfibM(k), sfibM(k - 1), sfibM(k - 2));
foreach (i, f; fibG(-9))
writef("%d:%d | ", i, f);
}


{{out}} for n = 85:

Fib( 85) =
D :   259695496911122586 <-   160500643816367088 +    99194853094755497
I :   259695496911122585 <-   160500643816367088 +    99194853094755497
O :   259695496911122585 <-   160500643816367088 +    99194853094755497
0:0 | -1:1 | -2:-1 | -3:2 | -4:-3 | -5:5 | -6:-8 | -7:13 | -8:-21 | -9:34 |


### Matrix Exponentiation Version

import std.bigint;

T fibonacciMatrix(T=BigInt)(size_t n) {
int[size_t.sizeof * 8] binDigits;
size_t nBinDigits;
while (n > 0) {
binDigits[nBinDigits] = n % 2;
n /= 2;
nBinDigits++;
}

T x=1, y, z=1;
foreach_reverse (b; binDigits[0 .. nBinDigits]) {
if (b) {
x = (x + z) * y;
y = y ^^ 2 + z ^^ 2;
} else {
auto x_old = x;
x = x ^^ 2 + y ^^ 2;
y = (x_old + z) * y;
}
z = x + y;
}

return y;
}

void main() {
10_000_000.fibonacciMatrix;
}


### Faster Version

For N = 10_000_000 this is about twice faster (run-time about 2.20 seconds) than the matrix exponentiation version.

import std.bigint, std.math;

// Algorithm from: Takahashi, Daisuke,
// "A fast algorithm for computing large Fibonacci numbers".
// Information Processing Letters 75.6 (30 November 2000): 243-246.
// Implementation from:
// pythonista.wordpress.com/2008/07/03/pure-python-fibonacci-numbers
BigInt fibonacci(in ulong n)
in {
assert(n > 0, "fibonacci(n): n must be > 0.");
} body {
if (n <= 2)
return 1.BigInt;
BigInt F = 1;
BigInt L = 1;
int sign = -1;
immutable uint n2 = cast(uint)n.log2.floor;
auto mask = 2.BigInt ^^ (n2 - 1);
foreach (immutable i; 1 .. n2) {
auto temp = F ^^ 2;
F = (F + L) / 2;
F = 2 * F ^^ 2 - 3 * temp - 2 * sign;
L = 5 * temp + 2 * sign;
sign = 1;
temp = F;
F = (F + L) / 2;
L = F + 2 * temp;
sign = -1;
}
}
if ((n & mask) == 0) {
F *= L;
} else {
F = (F + L) / 2;
F = F * L - sign;
}
return F;
}

void main() {
10_000_000.fibonacci;
}


## Dart

int fib(int n) {
if (n==0 || n==1) {
return n;
}
var prev=1;
var current=1;
for (var i=2; i<n; i++) {
var next = prev + current;
prev = current;
current = next;
}
return current;
}

int fibRec(int n) => n==0 || n==1 ? n : fibRec(n-1) + fibRec(n-2);

main() {
print(fib(11));
print(fibRec(11));
}


## Dc

This needs a modern Dc with r (swap) and # (comment). It easily can be adapted to an older Dc, but it will impact readability a lot.

[               # todo: n(<2) -- 1 and break 2 levels
d -           # 0
1 +           # 1
q
] s1

[               # todo: n(>-1) -- F(n)
d 0=1         # n(!=0)
d 1=1         # n(!in {0,1})
2 - d 1 +     # (n-2) (n-1)
lF x          # (n-2) F(n-1)
r             # F(n-1) (n-2)
lF x          # F(n-1)+F(n-2)
+
] sF

33 lF x f


{{out}}


5702887



## Delphi

### Iterative


function FibonacciI(N: Word): UInt64;
var
Last, New: UInt64;
I: Word;
begin
if N < 2 then
Result := N
else begin
Last := 0;
Result := 1;
for I := 2 to N do
begin
New := Last + Result;
Last := Result;
Result := New;
end;
end;
end;



### Recursive


function Fibonacci(N: Word): UInt64;
begin
if N < 2 then
Result := N
else
Result := Fibonacci(N - 1) + Fibonacci(N - 2);
end;



### Matrix

Algorithm is based on :$\begin\left\{pmatrix\right\}1&1\1&0\end\left\{pmatrix\right\}^n = \begin\left\{pmatrix\right\}F\left(n+1\right)&F\left(n\right)\F\left(n\right)&F\left(n-1\right)\end\left\{pmatrix\right\}$.


function fib(n: Int64): Int64;

type TFibMat = array[0..1] of array[0..1] of Int64;

function FibMatMul(a,b: TFibMat): TFibMat;
var i,j,k: integer;
tmp: TFibMat;
begin
for i := 0 to 1 do
for j := 0 to 1 do
begin
tmp[i,j] := 0;
for k := 0 to 1 do tmp[i,j] := tmp[i,j] + a[i,k] * b[k,j];
end;
FibMatMul := tmp;
end;

function FibMatExp(a: TFibMat; n: Int64): TFibmat;
begin
if n <= 1 then fibmatexp := a
else if (n mod 2 = 0) then FibMatExp := FibMatExp(FibMatMul(a,a), n div 2)
else if (n mod 2 = 1) then FibMatExp := FibMatMul(a, FibMatExp(FibMatMul(a,a), n div 2));
end;

var
matrix: TFibMat;

begin
matrix[0,0] := 1;
matrix[0,1] := 1;
matrix[1,0] := 1;
matrix[1,1] := 0;
if n > 1 then
matrix := fibmatexp(matrix,n-1);
fib := matrix[0,0];
end;



## DWScript

function fib(N : Integer) : Integer;
begin
if N < 2 then Result := 1
else Result := fib(N-2) + fib(N-1);
End;


## Dyalect

func fib(n) {
if n < 2 {
return n
} else {
return fib(n - 1) + fib(n - 2)
}
}

print(fib(30))


## E

def fib(n) {
var s := [0, 1]
for _ in 0..!n {
def [a, b] := s
s := [b, a+b]
}
return s[0]
}


(This version defines fib(0) = 0 because [http://www.research.att.com/~njas/sequences/A000045 OEIS A000045] does.)

## EasyLang

n = number input a = 0 b = 1 while n > 1 h = a + b a = b b = h n -= 1 . print b



## EchoLisp

Use '''memoization''' with the recursive version.

scheme

(define (fib n)
(if (< n 2) n
(+ (fib (- n 2)) (fib (1- n)))))

(remember 'fib #(0 1))

(for ((i 12)) (write (fib i)))
0 1 1 2 3 5 8 13 21 34 55 89



## ECL

### Analytic

//Calculates Fibonacci sequence up to n steps using Binet's closed form solution

FibFunction(UNSIGNED2 n) := FUNCTION
REAL Sqrt5 := Sqrt(5);
REAL Phi := (1+Sqrt(5))/2;
REAL Phi_Inv := 1/Phi;
UNSIGNED FibValue := ROUND( ( POWER(Phi,n)-POWER(Phi_Inv,n) ) /Sqrt5);
RETURN FibValue;
END;

FibSeries(UNSIGNED2 n) := FUNCTION

Fib_Layout := RECORD
UNSIGNED5 FibNum;
UNSIGNED5 FibValue;
END;

FibSeq := DATASET(n+1,
TRANSFORM
( Fib_Layout
, SELF.FibNum := COUNTER-1
, SELF.FibValue := IF(SELF.FibNum<2,SELF.FibNum, FibFunction(SELF.FibNum) )
)
);

RETURN FibSeq;

END; }


## EDSAC order code

This program calculates the nth—by default the tenth—number in the Fibonacci sequence and displays it (in binary) in the first word of storage tank 3.

[ Fibonacci sequence

### ============

A program for the EDSAC

Calculates the nth Fibonacci
number and displays it at the
top of storage tank 3

The default value of n is 10

To calculate other Fibonacci
numbers, set the starting value
of the count to n-2

Works with Initial Orders 2 ]

T56K  [ set load point  ]
GK    [ set theta       ]

[ Orders ]

[  0 ]  T20@  [ a = 0           ]
A17@  [ a += y          ]
U18@  [ temp = a        ]
A16@  [ a += x          ]
T17@  [ y = a; a = 0    ]
A18@  [ a += temp       ]
T16@  [ x = a; a = 0    ]

A19@  [ a = count       ]
S15@  [ a -= 1          ]
U19@  [ count = a       ]
E@    [ if a>=0 go to θ ]

T20@  [ a = 0           ]
A17@  [ a += y          ]
T96F  [ C(96) = a; a = 0]

ZF    [ halt ]

[ Data ]

[ 15 ]  P0D   [ const: 1        ]
[ 16 ]  P0F   [ var: x = 0      ]
[ 17 ]  P0D   [ var: y = 1      ]
[ 18 ]  P0F   [ var: temp = 0   ]
[ 19 ]  P4F   [ var: count = 8  ]
[ 20 ]  P0F   [ used to clear a ]

EZPF  [ begin execution ]


{{out}}

00000000000110111


## Eiffel


class
APPLICATION

create
make

feature

fibonacci (n: INTEGER): INTEGER
require
non_negative: n >= 0
local
i, n2, n1, tmp: INTEGER
do
n2 := 0
n1 := 1
from
i := 1
until
i >= n
loop
tmp := n1
n1 := n2 + n1
n2 := tmp
i := i + 1
end
Result := n1
if n = 0 then
Result := 0
end
end

feature {NONE} -- Initialization

make
-- Run application.
do
print (fibonacci (0))
print (" ")
print (fibonacci (1))
print (" ")
print (fibonacci (2))
print (" ")
print (fibonacci (3))
print (" ")
print (fibonacci (4))
print ("%N")
end

end



## Ela

Tail-recursive function:

fib = fib' 0 1
where fib' a b 0 = a
fib' a b n = fib' b (a + b) (n - 1)


Infinite (lazy) list:

fib = fib' 1 1
where fib' x y = & x :: fib' y (x + y)


## Elena

{{trans|Smalltalk}} ELENA 4.1 :

import extensions;

fibu(n)
{
int[] ac := new int[]::( 0,1 );
if (n < 2)
{
^ ac[n]
}
else
{
for(int i := 2, i <= n, i+=1)
{
int t := ac[1];
ac[1] := ac[0] + ac[1];
ac[0] := t
};

^ ac[1]
}
}

public program()
{
for(int i := 0, i <= 10, i+=1)
{
console.printLine(fibu(i))
}
}


{{out}}


0
1
1
2
3
5
8
13
21
34
55



## Elixir

defmodule Fibonacci do
def fib(0), do: 0
def fib(1), do: 1
def fib(n), do: fib(0, 1, n-2)

def fib(_, prv, -1), do: prv
def fib(prvprv, prv, n) do
next = prv + prvprv
fib(prv, next, n-1)
end
end

IO.inspect Enum.map(0..10, fn i-> Fibonacci.fib(i) end)


Using Stream:


Stream.unfold({0,1}, fn {a,b} -> {a,{b,a+b}} end) |> Enum.take(10)



{{out}}


[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]



## Elm

Naïve recursive implementation.

fibonacci : Int -> Int
fibonacci n = if n < 2 then
n
else
fibonacci(n - 2) + fibonacci(n - 1)


## Emacs Lisp

### version 1


(defun fib (n a b c)
(if (< c n) (fib n b (+ a b) (+ 1 c) )
(if (= c n) b a) ))

(defun fibonacci (n) (if (< n 2) n (fib n 0 1 1) ))



### version 2


(defun fibonacci (n)
(let ( (vec) (i) (j) (k) )
(if (< n 2) n
(progn
(setq vec (make-vector (+ n 1) 0) i 0 j 1 k 2)
(setf (aref vec 1) 1)
(while (<= k n)
(setf (aref vec k) (+ (elt vec i) (elt vec j) ))
(setq i (+ 1 i) j (+ 1 j) k (+ 1 k) ))
(elt vec n) ))))



Eval:


(insert
(mapconcat '(lambda (n) (format "%d" (fibonacci n) ))
(number-sequence 0 15) " ") )



Output:


0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610



## Erlang

### Recursive


-module(fib).
-export([fib/1]).

fib(0) -> 0;
fib(1) -> 1;
fib(N) -> fib(N-1) + fib(N-2).



### Iterative


-module(fiblin).
-export([fib/1])

fib(0) -> 0;
fib(1) -> 1;
fib(2) -> 1;
fib(3) -> 2;
fib(4) -> 3;
fib(5) -> 5;

fib(N) when is_integer(N) -> fib(N - 6, 5, 8).
fib(N, A, B) -> if N < 1 -> B; true -> fib(N-1, B, A+B) end.



Evaluate:


io:write([fiblin:fib(X) || X <- lists:seq(1,10) ]).



Output:


[1,1,2,3,5,8,13,21,34,55]ok



### Iterative 2


fib(N) -> fib(N, 0, 1).

fib(0, Result, _Next) -> Result;
fib(Iter, Result, Next) -> fib(Iter-1, Next, Result+Next).



## ERRE

!-------------------------------------------
! derived from my book "PROGRAMMARE IN ERRE"
! iterative solution
!-------------------------------------------

PROGRAM FIBONACCI

!$DOUBLE !VAR F1#,F2#,TEMP#,COUNT%,N% BEGIN !main INPUT("Number",N%) F1=0 F2=1 REPEAT TEMP=F2 F2=F1+F2 F1=TEMP COUNT%=COUNT%+1 UNTIL COUNT%=N% PRINT("FIB(";N%;")=";F2) ! Obviously a FOR loop or a WHILE loop can ! be used to solve this problem END PROGRAM  {{out}}  Number? 20 FIB( 20 )= 6765  ## Euphoria ==='Recursive' version=== {{works with|Euphoria|any version}}  function fibor(integer n) if n<2 then return n end if return fibor(n-1)+fibor(n-2) end function  ==='Iterative' version=== {{works with|Euphoria|any version}}  function fiboi(integer n) integer f0=0, f1=1, f if n<2 then return n end if for i=2 to n do f=f0+f1 f0=f1 f1=f end for return f end function  ==='Tail recursive' version=== {{works with|Euphoria|4.0.0}}  function fibot(integer n, integer u = 1, integer s = 0) if n < 1 then return s else return fibot(n-1,u+s,u) end if end function -- example: ? fibot(10) -- says 55  ==='Paper tape' version=== {{works with|Euphoria|4.0.0}}  include std/mathcons.e -- for PINF constant enum ADD, MOVE, GOTO, OUT, TEST, TRUETO global sequence tape = { 0, 1, { ADD, 2, 1 }, { TEST, 1, PINF }, { TRUETO, 0 }, { OUT, 1, "%.0f\n" }, { MOVE, 2, 1 }, { MOVE, 0, 2 }, { GOTO, 3 } } global integer ip global integer test global atom accum procedure eval( sequence cmd ) atom i = 1 while i <= length( cmd ) do switch cmd[ i ] do case ADD then accum = tape[ cmd[ i + 1 ] ] + tape[ cmd[ i + 2 ] ] i += 2 case OUT then printf( 1, cmd[ i + 2], tape[ cmd[ i + 1 ] ] ) i += 2 case MOVE then if cmd[ i + 1 ] = 0 then tape[ cmd[ i + 2 ] ] = accum else tape[ cmd[ i + 2 ] ] = tape[ cmd[ i + 1 ] ] end if i += 2 case GOTO then ip = cmd[ i + 1 ] - 1 -- due to ip += 1 in main loop i += 1 case TEST then if tape[ cmd[ i + 1 ] ] = cmd[ i + 2 ] then test = 1 else test = 0 end if i += 2 case TRUETO then if test then if cmd[ i + 1 ] = 0 then abort(0) else ip = cmd[ i + 1 ] - 1 end if end if end switch i += 1 end while end procedure test = 0 accum = 0 ip = 1 while 1 do -- embedded sequences (assumed to be code) are evaluated -- atoms (assumed to be data) are ignored if sequence( tape[ ip ] ) then eval( tape[ ip ] ) end if ip += 1 end while  ## FALSE [[$0=~][1-@@\$@@+\$44,.@]#]f:
20n: {First 20 numbers}
0 1 n;f;!%%44,. {Output: "0,1,1,2,3,5..."}


## Factor

### Iterative

: fib ( n -- m )
dup 2 < [
[ 0 1 ] dip [ swap [ + ] keep ] times
drop
] unless ;


### Recursive

: fib ( n -- m )
dup 2 < [
[ 1 - fib ] [ 2 - fib ] bi +
] unless ;


===Tail-Recursive===

: fib2 ( x y n -- a )
dup 1 <
[ 2drop ]
[ [ swap [ + ] keep ] dip 1 - fib2 ]
if ;
: fib ( n -- m ) [ 0 1 ] dip fib2 ;


### Matrix

{{trans|Ruby}}

USE: math.matrices

: fib ( n -- m )
dup 2 < [
[ { { 0 1 } { 1 1 } } ] dip 1 - m^n
second second
] unless ;


## Fancy

class Fixnum {
def fib {
match self -> {
case 0 -> 0
case 1 -> 1
case _ -> self - 1 fib + (self - 2 fib)
}
}
}

15 times: |x| {
x fib println
}



## Falcon

### Iterative

function fib_i(n)

if n < 2: return n

fibPrev = 1
fib = 1
for i in [2:n]
tmp = fib
fib += fibPrev
fibPrev = tmp
end
return fib
end


### Recursive

function fib_r(n)
if n < 2 :  return n
return fib_r(n-1) + fib_r(n-2)
end


### Tail Recursive

function fib_tr(n)
return fib_aux(n,0,1)
end
function fib_aux(n,a,b)
switch n
case 0 : return a
default: return fib_aux(n-1,a+b,a)
end
end


## Fantom

Ints have a limit of 64-bits, so overflow errors occur after computing Fib(92) = 7540113804746346429.


class Main
{
static Int fib (Int n)
{
if (n < 2) return n
fibNums := [1, 0]
while (fibNums.size <= n)
{
fibNums.insert (0, fibNums[0] + fibNums[1])
}
return fibNums.first
}

public static Void main ()
{
20.times |n|
{
echo ("Fib($n) is${fib(n)}")
}
}
}



## Fexl


# (fib n) = the nth Fibonacci number
\fib=
(
\loop==
(\x\y\n
le n 0 x;
\z=(+ x y)
\n=(- n 1)
loop y z n
)
loop 0 1
)

# Now test it:
for 0 20 (\n say (fib n))



{{out}}


0
1
1
2
3
5
8
13
21
34
55
89
144
233
377
610
987
1597
2584
4181
6765



## FOCAL

01.10 TYPE "FIBONACCI NUMBERS" !
01.30 SET A=0
01.40 SET B=1
01.50 FOR I=2,N; DO 2.0
01.60 TYPE "F(N) ", %8, B, !
01.70 QUIT

02.10 SET T=B
02.20 SET B=A+B
02.30 SET A=T


{{out}}

FIBONACCI NUMBERS
N =:20
F(N) =    6765


## Forth

: fib ( n -- fib )
0 1 rot 0 ?do  over + swap  loop drop ;


Since there are only a fixed and small amount of Fibonacci numbers that fit in a machine word, this FORTH version creates a table of Fibonacci numbers at compile time. It stops compiling numbers when there is arithmetic overflow (the number turns negative, indicating overflow.)

: F-start,  here 1 0 dup , ;
: F-next,   over + swap
dup 0> IF  dup , true  ELSE  false  THEN ;

: computed-table  ( compile: 'start 'next / run: i -- x )
create
>r execute
BEGIN  r@ execute not  UNTIL  rdrop
does>
swap cells + @ ;

' F-start, ' F-next,  computed-table fibonacci 2drop
here swap - cell/ Constant #F/64   \ # of fibonacci numbers generated

16 fibonacci . 987  ok
#F/64 . 93  ok
92 fibonacci . 7540113804746346429  ok   \ largest number generated.


## Fortran

### FORTRAN IV

C     FIBONACCI SEQUENCE - FORTRAN IV
NN=46
DO 1 I=0,NN
1 WRITE(*,300) I,IFIBO(I)
300 FORMAT(1X,I2,1X,I10)
END
C
FUNCTION IFIBO(N)
IF(N) 9,1,2
1 IFN=0
GOTO 9
2 IF(N-1) 9,3,4
3 IFN=1
GOTO 9
4 IFNM1=0
IFN=1
DO 5 I=2,N
IFNM2=IFNM1
IFNM1=IFN
5 IFN=IFNM1+IFNM2
9 IFIBO=IFN
END


{{out}}


0          0
1          1
2          1
3          2
4          3
5          5
6          8
7         13
8         21
9         34
10         55
...
45 1134903170
46 1836311903



### FORTRAN 77


FUNCTION IFIB(N)
IF (N.EQ.0) THEN
ITEMP0=0
ELSE IF (N.EQ.1) THEN
ITEMP0=1
ELSE IF (N.GT.1) THEN
ITEMP1=0
ITEMP0=1
DO 1 I=2,N
ITEMP2=ITEMP1
ITEMP1=ITEMP0
ITEMP0=ITEMP1+ITEMP2
1   CONTINUE
ELSE
ITEMP1=1
ITEMP0=0
DO 2 I=-1,N,-1
ITEMP2=ITEMP1
ITEMP1=ITEMP0
ITEMP0=ITEMP2-ITEMP1
2   CONTINUE
END IF
IFIB=ITEMP0
END



Test program


EXTERNAL IFIB
CHARACTER*10 LINE
PARAMETER ( LINE = '----------' )
WRITE(*,900) 'N', 'F[N]', 'F[-N]'
WRITE(*,900) LINE, LINE, LINE
DO 1 N = 0, 10
WRITE(*,901) N, IFIB(N), IFIB(-N)
1 CONTINUE
900 FORMAT(3(X,A10))
901 FORMAT(3(X,I10))
END



{{out}}


N       F[N]      F[-N]
---------- ---------- ----------
0          0          0
1          1          1
2          1         -1
3          2          2
4          3         -3
5          5          5
6          8         -8
7         13         13
8         21        -21
9         34         34
10         55        -55



### Recursive

In ISO Fortran 90 or later, use a RECURSIVE function:

module fibonacci
contains
recursive function fibR(n) result(fib)
integer, intent(in) :: n
integer             :: fib

select case (n)
case (:0);      fib = 0
case (1);       fib = 1
case default;   fib = fibR(n-1) + fibR(n-2)
end select
end function fibR


### Iterative

In ISO Fortran 90 or later:

    function fibI(n)
integer, intent(in) :: n
integer, parameter :: fib0 = 0, fib1 = 1
integer            :: fibI, back1, back2, i

select case (n)
case (:0);      fibI = fib0
case (1);       fibI = fib1

case default
fibI = fib1
back1 = fib0
do i = 2, n
back2 = back1
back1 = fibI
fibI   = back1 + back2
end do
end select
end function fibI
end module fibonacci


Test program

program fibTest
use fibonacci

do i = 0, 10
print *, fibr(i), fibi(i)
end do
end program fibTest


{{out}}


0 0
1 1
1 1
2 2
3 3
5 5
8 8
13 13
21 21
34 34
55 55



## FreeBASIC

Extended sequence coded big integer.

'Fibonacci extended
'Freebasic version 24  Windows
Dim Shared ADDQmod(0 To 19) As Ubyte
Dim Shared ADDbool(0 To 19) As Ubyte

For z As Integer=0 To 19
Next z

Function plusINT(NUM1 As String,NUM2 As String) As String
Dim As Byte flag
#macro finish()
three=Ltrim(three,"0")
If three="" Then Return "0"
If flag=1 Then Swap NUM2,NUM1
Return three
Exit Function
#endmacro
var lenf=Len(NUM1)
var lens=Len(NUM2)
If lens>lenf Then
Swap NUM2,NUM1
Swap lens,lenf
flag=1
End If

var diff=lenf-lens-Sgn(lenf-lens)
var three="0"+NUM1
var two=String(lenf-lens,"0")+NUM2
Dim As Integer n2

For n2=lenf-1 To diff Step -1
Next n2
finish()
End If
If n2=-1 Then
finish()
End If

For n2=n2 To 0 Step -1
Next n2
finish()
End Function

Function  fibonacci(n As Integer) As String
Dim As String sl,l,term
sl="0": l="1"
If n=1 Then Return "0"
If n=2 Then Return "1"
n=n-2
For x As Integer= 1 To n
term=plusINT(l,sl)
sl=l
l=term
Next x
Function =term
End Function

'
### ===========  EXAMPLE ============

print "THE SEQUENCE TO 10:"
print
For n As Integer=1 To 10
Print "term";n;": "; fibonacci(n)
Next n
print
print "Selected Fibonacci number"
print "Fibonacci 500"
print
print fibonacci(500)
Sleep


{{out}}

THE SEQUENCE TO 10:

term 1: 0
term 2: 1
term 3: 1
term 4: 2
term 5: 3
term 6: 5
term 7: 8
term 8: 13
term 9: 21
term 10: 34

Selected Fibonacci number
Fibonacci 500

86168291600238450732788312165664788095941068326060883324529903470149056115823592
713458328176574447204501


## Free Pascal

type
/// domain for Fibonacci function
/// where result is within nativeUInt
// You can not name it fibonacciDomain,
// since the Fibonacci function itself
// is defined for all whole numbers
// but the result beyond F(n) exceeds high(nativeUInt).
fibonacciLeftInverseRange =
{$ifdef CPU64} 0..93 {$else} 0..47 {$endif}; {** implements Fibonacci sequence iteratively \param n the index of the Fibonacci number to calculate \returns the Fibonacci value at n } function fibonacci(const n: fibonacciLeftInverseRange): nativeUInt; type /// more meaningful identifiers than simple integers relativePosition = (previous, current, next); var /// temporary iterator variable i: longword; /// holds preceding fibonacci values f: array[relativePosition] of nativeUInt; begin f[previous] := 0; f[current] := 1; // note, in Pascal for-loop-limits are inclusive for i := 1 to n do begin f[next] := f[previous] + f[current]; f[previous] := f[current]; f[current] := f[next]; end; // assign to previous, bc f[current] = f[next] for next iteration fibonacci := f[previous]; end;  ## Frink All of Frink's integers can be arbitrarily large.  fibonacciN[n] := { a = 0 b = 1 count = 0 while count < n { [a,b] = [b, a + b] count = count + 1 } return a }  ## FRISC Assembly To find the nth Fibonacci number, call this subroutine with n in register R0: the answer will be returned in R0 too. Contents of other registers are preserved. FIBONACCI PUSH R1 PUSH R2 PUSH R3 MOVE 0, R1 MOVE 1, R2 FIB_LOOP SUB R0, 1, R0 JP_Z FIB_DONE MOVE R2, R3 ADD R1, R2, R2 MOVE R3, R1 JP FIB_LOOP FIB_DONE MOVE R2, R0 POP R3 POP R2 POP R1 RET  =={{header|F_Sharp|F#}}== This is a fast [tail-recursive] approach using the F# big integer support:  let fibonacci n : bigint = let rec f a b n = match n with | 0 -> a | 1 -> b | n -> (f b (a + b) (n - 1)) f (bigint 0) (bigint 1) n > fibonacci 100;; val it : bigint = 354224848179261915075I  Lazy evaluated using sequence workflow: let rec fib = seq { yield! [0;1]; for (a,b) in Seq.zip fib (Seq.skip 1 fib) -> a+b}  The above is extremely slow due to the nested recursions on sequences, which aren't very efficient at the best of times. The above takes seconds just to compute the 30th Fibonacci number! Lazy evaluation using the sequence unfold anamorphism is much much better as to efficiency: let fibonacci = Seq.unfold (fun (x, y) -> Some(x, (y, x + y))) (0I,1I) fibonacci |> Seq.nth 10000  Approach similar to the Matrix algorithm in C#, with some shortcuts involved. Since it uses exponentiation by squaring, calculations of fib(n) where n is a power of 2 are particularly quick. Eg. fib(2^20) was calculated in a little over 4 seconds on this poster's laptop.  open System open System.Diagnostics open System.Numerics /// Finds the highest power of two which is less than or equal to a given input. let inline prevPowTwo (x : int) = let mutable n = x n <- n - 1 n <- n ||| (n >>> 1) n <- n ||| (n >>> 2) n <- n ||| (n >>> 4) n <- n ||| (n >>> 8) n <- n ||| (n >>> 16) n <- n + 1 match x with | x when x = n -> x | _ -> n/2 /// Evaluates the nth Fibonacci number using matrix arithmetic and /// exponentiation by squaring. let crazyFib (n : int) = let powTwo = prevPowTwo n /// Applies 2n rule repeatedly until another application of the rule would /// go over the target value (or the target value has been reached). let rec iter1 i q r s = match i with | i when i < powTwo -> iter1 (i*2) (q*q + r*r) (r * (q+s)) (r*r + s*s) | _ -> i, q, r, s /// Applies n+1 rule until the target value is reached. let rec iter2 (i, q, r, s) = match i with | i when i < n -> iter2 ((i+1), (q+r), q, r) | _ -> q match n with | 0 -> 1I | _ -> iter1 1 1I 1I 0I |> iter2  ## FunL ### Recursive def fib( 0 ) = 0 fib( 1 ) = 1 fib( n ) = fib( n - 1 ) + fib( n - 2 )  ### Tail Recursive def fib( n ) = def _fib( 0, prev, _ ) = prev _fib( 1, _, next ) = next _fib( n, prev, next ) = _fib( n - 1, next, next + prev ) _fib( n, 0, 1 )  ### Lazy List val fib = def _fib( a, b ) = a # _fib( b, a + b ) _fib( 0, 1 ) println( fib(10000) )  {{out}}  33644764876431783266621612005107543310302148460680063906564769974680081442166662368155595513633734025582065332680836159373734790483865268263040892463056431887354544369559827491606602099884183933864652731300088830269235673613135117579297437854413752130520504347701602264758318906527890855154366159582987279682987510631200575428783453215515103870818298969791613127856265033195487140214287532698187962046936097879900350962302291026368131493195275630227837628441540360584402572114334961180023091208287046088923962328835461505776583271252546093591128203925285393434620904245248929403901706233888991085841065183173360437470737908552631764325733993712871937587746897479926305837065742830161637408969178426378624212835258112820516370298089332099905707920064367426202389783111470054074998459250360633560933883831923386783056136435351892133279732908133732642652633989763922723407882928177953580570993691049175470808931841056146322338217465637321248226383092103297701648054726243842374862411453093812206564914032751086643394517512161526545361333111314042436854805106765843493523836959653428071768775328348234345557366719731392746273629108210679280784718035329131176778924659089938635459327894523777674406192240337638674004021330343297496902028328145933418826817683893072003634795623117103101291953169794607632737589253530772552375943788434504067715555779056450443016640119462580972216729758615026968443146952034614932291105970676243268515992834709891284706740862008587135016260312071903172086094081298321581077282076353186624611278245537208532365305775956430072517744315051539600905168603220349163222640885248852433158051534849622434848299380905070483482449327453732624567755879089187190803662058009594743150052402532709746995318770724376825907419939632265984147498193609285223945039707165443156421328157688908058783183404917434556270520223564846495196112460268313970975069382648706613264507665074611512677522748621598642530711298441182622661057163515069260029861704945425047491378115154139941550671256271197133252763631939606902895650288268608362241082050562430701794976171121233066073310059947366875  ### Iterative def fib( n ) = a, b = 0, 1 for i <- 1..n a, b = b, a+b a  === Binet's Formula === import math.sqrt def fib( n ) = phi = (1 + sqrt( 5 ))/2 int( (phi^n - (-phi)^-n)/sqrt(5) + .5 )  ### Matrix Exponentiation def mul( a, b ) = res = array( a.length(), b(0).length() ) for i <- 0:a.length(), j <- 0:b(0).length() res( i, j ) = sum( a(i, k)*b(k, j) | k <- 0:b.length() ) vector( res ) def pow( _, 0 ) = ((1, 0), (0, 1)) pow( x, 1 ) = x pow( x, n ) | 2|n = pow( mul(x, x), n\2 ) | otherwise = mul(x, pow( mul(x, x), (n - 1)\2 ) ) def fib( n ) = pow( ((0, 1), (1, 1)), n )(0, 1) for i <- 0..10 println( fib(i) )  {{out}}  0 1 1 2 3 5 8 13 21 34 55  ## Futhark ### Iterative  fun main(n: int): int = loop((a,b) = (0,1)) = for _i < n do (b, a + b) in a  ## FutureBasic ### Iterative  include "Tlbx Timer.incl" include "ConsoleWindow" local fn Fibonacci( n as long ) as Long begin globals dim as long s1, s2// static end globals dim as long temp if ( n < 2 ) s1 = n exit fn else temp = s1 + s2 s2 = s1 s1 = temp exit fn end if end fn = s1 dim as long i dim as UnsignedWide start, finish Microseconds( @start ) for i = 0 to 40 print i; ". "; fn Fibonacci(i) next i Microseconds( @finish ) print "Compute time:"; (finish.lo - start.lo ) / 1000; " ms"  Output:  0. 0 1. 1 2. 1 3. 2 4. 3 5. 5 6. 8 7. 13 8. 21 9. 34 10. 55 11. 89 12. 144 13. 233 14. 377 15. 610 16. 987 17. 1597 18. 2584 19. 4181 20. 6765 21. 10946 22. 17711 23. 28657 24. 46368 25. 75025 26. 121393 27. 196418 28. 317811 29. 514229 30. 832040 31. 1346269 32. 2178309 33. 3524578 34. 5702887 35. 9227465 36. 14930352 37. 24157817 38. 39088169 39. 63245986 40. 102334155 41. 165580141 42. 267914296 43. 433494437 44. 701408733 45. 1134903170 46. 1836311903 47. 2971215073 48. 4.80752698e+9 49. 7.77874205e+9 50. 1.2586269e+10 51. 2.03650111e+10 52. 3.29512801e+10 53. 5.33162912e+10 54. 8.62675713e+10 55. 1.39583862e+11 56. 2.25851434e+11 57. 3.65435296e+11 58. 5.9128673e+11 59. 9.56722026e+11 60. 1.54800876e+12 61. 2.50473078e+12 62. 4.05273954e+12 63. 6.55747032e+12 64. 1.06102099e+13 65. 1.71676802e+13 66. 2.777789e+13 67. 4.49455702e+13 68. 7.27234602e+13 69. 1.1766903e+14 70. 1.90392491e+14 71. 3.08061521e+14 72. 4.98454012e+14 73. 8.06515533e+14 74. 1.30496954e+15 75. 2.11148508e+15 76. 3.41645462e+15 77. 5.5279397e+15 78. 8.94439432e+15 79. 1.4472334e+16 80. 2.34167283e+16 81. 3.78890624e+16 82. 6.13057907e+16 83. 9.91948531e+16 84. 1.60500644e+17 85. 2.59695497e+17 86. 4.20196141e+17 87. 6.79891638e+17 88. 1.10008778e+18 89. 1.77997942e+18 90. 2.88006719e+18 91. 4.66004661e+18 92. 7.5401138e+18 93. 1.22001604e+19 94. 1.97402742e+19 95. 3.19404346e+19 96. 5.16807089e+19 97. 8.36211435e+19 98. 1.35301852e+20 99. 2.18922996e+20 100. 3.54224848e+20 Compute time: 15 ms  ## GAP fib := function(n) local a; a := [[0, 1], [1, 1]]^n; return a[1][2]; end;  GAP has also a buit-in function for that. Fibonacci(n);  ## Gecho  Prints the first several fibonacci numbers... ## GFA Basic <lang> ' ' Compute nth Fibonacci number ' ' open a window for display OPENW 1 CLEARW 1 ' Display some fibonacci numbers ' Fib(46) is the largest number GFA Basic can reach ' (long integers are 4 bytes) FOR i%=0 TO 46 PRINT "fib(";i%;")=";@fib(i%) NEXT i% ' wait for a key press and tidy up ~INP(2) CLOSEW 1 ' ' Function to compute nth fibonacci number ' n must be in range 0 to 46, inclusive ' FUNCTION fib(n%) LOCAL n0%,n1%,nn%,i% n0%=0 n1%=1 SELECT n% CASE 0 RETURN n0% CASE 1 RETURN n1% DEFAULT FOR i%=2 TO n% nn%=n0%+n1% n0%=n1% n1%=nn% NEXT i% RETURN nn% ENDSELECT ENDFUNC  ## GML ///fibonacci(n) //Returns the nth fibonacci number var n, numb; n = argument0; if (n == 0) { numb = 0; } else { var fm2, fm1; fm2 = 0; fm1 = 1; numb = 1; repeat(n-1) { numb = fm2+fm1; fm2 = fm1; fm1 = numb; } } return numb;  ## Go ### Recursive func fib(a int) int { if a < 2 { return a } return fib(a - 1) + fib(a - 2) }  ### Iterative import ( "math/big" ) func fib(n uint64) *big.Int { if n < 2 { return big.NewInt(int64(n)) } a, b := big.NewInt(0), big.NewInt(1) for n--; n > 0; n-- { a.Add(a, b) a, b = b, a } return b }  ### Iterative using a closure func fibNumber() func() int { fib1, fib2 := 0, 1 return func() int { fib1, fib2 = fib2, fib1 + fib2 return fib1 } } func fibSequence(n int) int { f := fibNumber() fib := 0 for i := 0; i < n; i++ { fib = f() } return fib }  ### Using a goroutine and channel func fib(c chan int) { a, b := 0, 1 for { c <- a a, b = b, a+b } } func main() { c := make(chan int) go fib(c) for i := 0; i < 10; i++ { fmt.Println(<-c) } }  ## Groovy Full "extra credit" solutions. ### Recursive A recursive closure must be ''pre-declared''. def rFib rFib = { it == 0 ? 0 : it == 1 ? 1 : it > 1 ? rFib(it-1) + rFib(it-2) /*it < 0*/: rFib(it+2) - rFib(it+1) }  ### Iterative def iFib = { it == 0 ? 0 : it == 1 ? 1 : it > 1 ? (2..it).inject([0,1]){i, j -> [i[1], i[0]+i[1]]}[1] /*it < 0*/: (-1..it).inject([0,1]){i, j -> [i[1]-i[0], i[0]]}[0] }  ### Analytic final φ = (1 + 5**(1/2))/2 def aFib = { (φ**it - (-φ)**(-it))/(5**(1/2)) as BigInteger }  Test program: def time = { Closure c -> def start = System.currentTimeMillis() def result = c() def elapsedMS = (System.currentTimeMillis() - start)/1000 printf '(%6.4fs elapsed)', elapsedMS result } print " F(n) elapsed time "; (-10..10).each { printf ' %3d', it }; println() print "--------- -----------------"; (-10..10).each { print ' ---' }; println() [recursive:rFib, iterative:iFib, analytic:aFib].each { name, fib -> printf "%9s ", name def fibList = time { (-10..10).collect {fib(it)} } fibList.each { printf ' %3d', it } println() }  {{out}}  F(n) elapsed time -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 --------- ----------------- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- recursive (0.0080s elapsed) -55 34 -21 13 -8 5 -3 2 -1 1 0 1 1 2 3 5 8 13 21 34 55 iterative (0.0040s elapsed) -55 34 -21 13 -8 5 -3 2 -1 1 0 1 1 2 3 5 8 13 21 34 55 analytic (0.0030s elapsed) -55 34 -21 13 -8 5 -3 2 -1 1 0 1 1 2 3 5 8 13 21 34 55  ## Harbour ### Recursive  #include "harbour.ch" Function fibb(a,b,n) return(if(--n>0,fibb(b,a+b,n),a))  ### Iterative  #include "harbour.ch" Function fibb(n) local fnow:=0, fnext:=1, tempf while (--n>0) tempf:=fnow+fnext fnow:=fnext fnext:=tempf end while return(fnext)  ## Haskell ### Analytic main :: IO () main = print [ floor (0.01 + (1 / p ** n + p ** n) / sqrt 5) | let p = (1 + sqrt 5) / 2 , n <- [0 .. 42] ]  ### Recursive Simple definition, very inefficient. fib x = if x < 1 then 0 else if x < 2 then 1 else fib (x - 1) + fib (x - 2)  ### Recursive with Memoization Very fast. fib x = if x < 1 then 0 else if x == 1 then 1 else fibs !! (x - 1) + fibs !! (x - 2) where fibs = map fib [0 ..]  ### Recursive with Memoization using memoized library Even faster and simpler is to use a defined memoizer (e.g. from MemoTrie package): import Data.MemoTrie fib :: Integer -> Integer fib = memo f where f 0 = 0 f 1 = 1 f n = fib (n-1) + fib (n-2)  You can rewrite this without introducing f explicitly import Data.MemoTrie fib :: Integer -> Integer fib = memo$ \x -> case x of
0 -> 0
1 -> 1
n -> fib (n-1) + fib (n-2)


Or using LambdaCase extension you can write it even shorter:

{-# Language LambdaCase #-}
import Data.MemoTrie
fib :: Integer -> Integer
fib = memo $\case 0 -> 0 1 -> 1 n -> fib (n-1) + fib (n-2)  The version that supports negative numbers: {-# Language LambdaCase #-} import Data.MemoTrie fib :: Integer -> Integer fib = memo$ \case
0 -> 0
1 -> 1
n | n>0 -> fib (n-1) + fib (n-2)
| otherwise -> fib (n+2) - fib (n+1)


### Iterative

fib n = go n 0 1
where
go n a b
| n == 0 = a
| otherwise = go (n - 1) b (a + b)


### = With lazy lists =

This is a standard example how to use lazy lists. Here's the (infinite) list of all Fibonacci numbers:

fib = 0 : 1 : zipWith (+) fib (tail fib)


Or alternatively:

fib = 0 : 1 : (zipWith (+) <*> tail) fib


The ''n''th Fibonacci number is then just fib !! n. The above is equivalent to

fib = 0 : 1 : next fib where next (a: t@(b:_)) = (a+b) : next t


Also

fib = 0 : scanl (+) 1 fib


### = As a fold =

Accumulator holds last two members of the series:

import Data.List (foldl') --'

fib :: Integer -> Integer
fib n =
fst $foldl' --' ($$a, b) _ -> (b, a + b)) (0, 1) [1 .. n]  ### With matrix exponentiation Adapting the (rather slow) code from [[Matrix exponentiation operator]], we can simply write: import Data.List (transpose) fib :: (Integral b, Num a) => b -> a fib 0 = 0 -- this line is necessary because "something ^ 0" returns "fromInteger 1", which unfortunately -- in our case is not our multiplicative identity (the identity matrix) but just a 1x1 matrix of 1 fib n = (last . head . unMat) (Mat [[1, 1], [1, 0]] ^ n) -- Code adapted from Matrix exponentiation operator task --------------------- (<+>) :: Num c => [c] -> [c] -> [c] (<+>) = zipWith (+) (<*>) :: Num a => [a] -> [a] -> a (<*>) = (sum .) . zipWith (*) newtype Mat a = Mat { unMat :: [[a]] } deriving (Eq) instance Show a => Show (Mat a) where show xm = "Mat " ++ show (unMat xm) instance Num a => Num (Mat a) where negate xm = Mat map (map negate) unMat xm xm + ym = Mat zipWith (<+>) (unMat xm) (unMat ym) xm * ym = Mat [ [ xs Main.<*> ys -- to distinguish from standard applicative operator | ys <- transpose unMat ym ] | xs <- unMat xm ] fromInteger n = Mat [[fromInteger n]] abs = undefined signum = undefined -- TEST ---------------------------------------------------------------------- main :: IO () main = (print . take 10 . show . fib) (10 ^ 5)  So, for example, the hundred-thousandth Fibonacci number starts with the digits: {{Out}} "2597406934"  ### With recurrence relations Using Fib[m=3n+r] [http://en.wikipedia.org/wiki/Fibonacci_number#Other_identities recurrence identities]: import Control.Arrow ((&&&)) fibstep :: (Integer, Integer) -> (Integer, Integer) fibstep (a, b) = (b, a + b) fibnums :: [Integer] fibnums = map fst iterate fibstep (0, 1) fibN2 :: Integer -> (Integer, Integer) fibN2 m | m < 10 = iterate fibstep (0, 1) !! fromIntegral m fibN2 m = fibN2_next (n, r) (fibN2 n) where (n, r) = quotRem m 3 fibN2_next (n, r) (f, g) | r == 0 = (a, b) -- 3n ,3n+1 | r == 1 = (b, c) -- 3n+1,3n+2 | r == 2 = (c, d) -- 3n+2,3n+3 (*) where a = 5 * f ^ 3 + if even n then 3 * f else (-3 * f) -- 3n b = g ^ 3 + 3 * g * f ^ 2 - f ^ 3 -- 3n+1 c = g ^ 3 + 3 * g ^ 2 * f + f ^ 3 -- 3n+2 d = 5 * g ^ 3 + if even n then (-3 * g) else 3 * g -- 3(n+1) (*) main :: IO () main = print (length &&& take 20) . show . fst fibN2 (10 ^ 2)  {{Out}} (21,"35422484817926191507")  (fibN2 n) directly calculates a pair (f,g) of two consecutive Fibonacci numbers, (Fib[n], Fib[n+1]), from recursively calculated such pair at about n/3:  *Main> (length &&& take 20) . show . fst fibN2 (10^6) (208988,"19532821287077577316")  The above should take less than 0.1s to calculate on a modern box. Other identities that could also be used are [http://en.wikipedia.org/wiki/Fibonacci_number#Matrix_form here]. In particular, for (n-1,n) ---> (2n-1,2n) transition which is equivalent to the matrix exponentiation scheme, we have f (n,(a,b)) = (2*n,(a*a+b*b,2*a*b+b*b)) -- iterate f (1,(0,1)) ; b is nth  and for (n,n+1) ---> (2n,2n+1) (derived from d'Ocagne's identity, for example), g (n,(a,b)) = (2*n,(2*a*b-a*a,a*a+b*b)) -- iterate g (1,(1,1)) ; a is nth  ## Haxe ### Iterative static function fib(steps:Int, handler:Int->Void) { var current = 0; var next = 1; for (i in 1...steps) { handler(current); var temp = current + next; current = next; next = temp; } handler(current); }  ### As Iterator class FibIter { private var current = 0; private var nextItem = 1; private var limit:Int; public function new(limit) this.limit = limit; public function hasNext() return limit > 0; public function next() { limit--; var ret = current; var temp = current + nextItem; current = nextItem; nextItem = temp; return ret; } }  Used like: for (i in new FibIter(10)) Sys.println(i);  ## Hope ### Recursive dec f : num -> num; --- f 0 <= 0; --- f 1 <= 1; --- f(n+2) <= f n + f(n+1);  ===Tail-recursive=== dec fib : num -> num; --- fib n <= l (1, 0, n) whererec l == \(a,b,succ c) => if c<1 then a else l((a+b),a,c) |(a,b,0) => 0;  ### With lazy lists This language, being one of Haskell's ancestors, also has lazy lists. Here's the (infinite) list of all Fibonacci numbers: dec fibs : list num; --- fibs <= fs whererec fs == 0::1::map (+) (tail fs||fs);  The ''n''th Fibonacci number is then just fibs @ n. ## HicEst REAL :: Fibonacci(10) Fibonacci = (==2) + Fibonacci(-1) + Fibonacci(-2) WRITE(ClipBoard) Fibonacci ! 0 1 1 2 3 5 8 13 21 34  ## Hy Recursive implementation. (defn fib [n] (if (< n 2) n (+ (fib (- n 2)) (fib (- n 1)))))  =={{header|Icon}} and {{header|Unicon}}== Icon has built-in support for big numbers. First, a simple recursive solution augmented by caching for non-negative input. This examples computes fib(1000) if there is no integer argument. procedure main(args) write(fib(integer(!args) | 1000) end procedure fib(n) static fCache initial { fCache := table() fCache[0] := 0 fCache[1] := 1 } /fCache[n] := fib(n-1) + fib(n-2) return fCache[n] end  {{libheader|Icon Programming Library}} The above solution is similar to the one provided [http://www.cs.arizona.edu/icon/library/src/procs/memrfncs.icn fib in memrfncs] Now, an O(logN) solution. For large N, it takes far longer to convert the result to a string for output than to do the actual computation. This example computes fib(1000000) if there is no integer argument. procedure main(args) write(fib(integer(!args) | 1000000)) end procedure fib(n) return fibMat(n)[1] end procedure fibMat(n) if n <= 0 then return [0,0] if n = 1 then return [1,0] fp := fibMat(n/2) c := fp[1]*fp[1] + fp[2]*fp[2] d := fp[1]*(fp[1]+2*fp[2]) if n%2 = 1 then return [c+d, d] else return [d, c] end  ## IDL ### Recursive function fib,n if n lt 3 then return,1L else return, fib(n-1)+fib(n-2) end  Execution time O(2^n) until memory is exhausted and your machine starts swapping. Around fib(35) on a 2GB Core2Duo. ### Iterative function fib,n psum = (csum = 1uL) if n lt 3 then return,csum for i = 3,n do begin nsum = psum + csum psum = csum csum = nsum endfor return,nsum end  Execution time O(n). Limited by size of uLong to fib(49) ### Analytic function fib,n q=1/( p=(1+sqrt(5))/2 ) return,round((p^n+q^n)/sqrt(5)) end  Execution time O(1), only limited by the range of LongInts to fib(48). ## Idris ### Analytic fibAnalytic : Nat -> Double fibAnalytic n = floor ((pow goldenRatio n) - (pow (-1.0/goldenRatio) n)) / sqrt(5) where goldenRatio : Double goldenRatio = (1.0 + sqrt(5)) / 2.0  ### Recursive fibRecursive : Nat -> Nat fibRecursive Z = Z fibRecursive (S Z) = (S Z) fibRecursive (S (S n)) = fibRecursive (S n) + fibRecursive n  ### Iterative fibIterative : Nat -> Nat fibIterative n = fibIterative' n Z (S Z) where fibIterative' : Nat -> Nat -> Nat -> Nat fibIterative' Z a _ = a fibIterative' (S n) a b = fibIterative' n b (a + b)  ### Lazy fibLazy : Lazy (List Nat) fibLazy = 0 :: 1 :: zipWith (+) fibLazy ( case fibLazy of (x::xs) => xs [] => [])  ## J The [[j:Essays/Fibonacci_Sequence|Fibonacci Sequence essay]] on the J Wiki presents a number of different ways of obtaining the nth Fibonacci number. Here is one:  fibN=: (-&2 +&: -&1)^:(1&<) M."0  '''Examples:'''  fibN 12 144 fibN i.31 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040  (This implementation is doubly recursive except that results are cached across function calls.) ## Java ### Iterative public static long itFibN(int n) { if (n < 2) return n; long ans = 0; long n1 = 0; long n2 = 1; for(n--; n > 0; n--) { ans = n1 + n2; n1 = n2; n2 = ans; } return ans; }   /** * O(log(n)) */ public static long fib(long n) { if (n <= 0) return 0; long i = (int) (n - 1); long a = 1, b = 0, c = 0, d = 1, tmp1,tmp2; while (i > 0) { if (i % 2 != 0) { tmp1 = d * b + c * a; tmp2 = d * (b + a) + c * b; a = tmp1; b = tmp2; } tmp1 = (long) (Math.pow(c, 2) + Math.pow(d, 2)); tmp2 = d * (2 * c + d); c = tmp1; d = tmp2; i = i / 2; } return a + b; }  ### Recursive public static long recFibN(final int n) { return (n < 2) ? n : recFibN(n - 1) + recFibN(n - 2); }  ### Analytic This method works up to the 92nd Fibonacci number. After that, it goes out of range. public static long anFibN(final long n) { double p = (1 + Math.sqrt(5)) / 2; double q = 1 / p; return (long) ((Math.pow(p, n) + Math.pow(q, n)) / Math.sqrt(5)); }  ===Tail-recursive=== public static long fibTailRec(final int n) { return fibInner(0, 1, n); } private static long fibInner(final long a, final long b, final int n) { return n < 1 ? a : n == 1 ? b : fibInner(b, a + b, n - 1); }  ### Streams  import java.util.function.LongUnaryOperator; import java.util.stream.LongStream; public class FibUtil { public static LongStream fibStream() { return LongStream.iterate( 1l, new LongUnaryOperator() { private long lastFib = 0; @Override public long applyAsLong( long operand ) { long ret = operand + lastFib; lastFib = operand; return ret; } }); } public static long fib(long n) { return fibStream().limit( n ).reduce((prev, last) -> last).getAsLong(); } }  ## JavaScript ### ES5 ### =Recursive= Basic recursive function: function fib(n) { return n<2?n:fib(n-1)+fib(n-2); }  Can be rewritten as: function fib(n) { if (n<2) { return n; } else { return fib(n-1)+fib(n-2); } }  One possibility familiar to Scheme programmers is to define an internal function for iteration through anonymous tail recursion: function fib(n) { return function(n,a,b) { return n>0 ? arguments.callee(n-1,b,a+b) : a; }(n,0,1); }  ### Iterative function fib(n) { var a = 0, b = 1, t; while (n-- > 0) { t = a; a = b; b += t; console.log(a); } return a; }  ### =Memoization= With the keys of a dictionary, var fib = (function(cache){ return cache = cache || {}, function(n){ if (cache[n]) return cache[n]; else return cache[n] = n == 0 ? 0 : n < 0 ? -fib(-n) : n <= 2 ? 1 : fib(n-2) + fib(n-1); }; })();  with the indices of an array, (function () { 'use strict'; function fib(n) { return Array.apply(null, Array(n + 1)) .map(function (_, i, lst) { return lst[i] = ( i ? i < 2 ? 1 : lst[i - 2] + lst[i - 1] : 0 ); })[n]; } return fib(32); })();  {{Out}} 2178309  ====Y-Combinator==== function Y(dn) { return (function(fn) { return fn(fn); }(function(fn) { return dn(function() { return fn(fn).apply(null, arguments); }); })); } var fib = Y(function(fn) { return function(n) { if (n === 0 || n === 1) { return n; } return fn(n - 1) + fn(n - 2); }; });  ### =Generators= function* fibonacciGenerator() { var prev = 0; var curr = 1; while (true) { yield curr; curr = curr + prev; prev = curr - prev; } } var fib = fibonacciGenerator();  ### ES6 ### =Memoized= If we want access to the whole preceding series, as well as a memoized route to a particular member, we can use an accumulating fold. (() => { 'use strict'; // Nth member of fibonacci series // fib :: Int -> Int function fib(n) { return mapAccumL(([a, b]) => [ [b, a + b], b ], [0, 1], range(1, n))[0][0]; }; // GENERIC FUNCTIONS // mapAccumL :: (acc -> x -> (acc, y)) -> acc -> [x] -> (acc, [y]) let mapAccumL = (f, acc, xs) => { return xs.reduce((a, x) => { let pair = f(a[0], x); return [pair[0], a[1].concat(pair[1])]; }, [acc, []]); } // range :: Int -> Int -> Maybe Int -> [Int] let range = (m, n) => Array.from({ length: Math.floor(n - m) + 1 }, (_, i) => m + i); // TEST return fib(32); // --> 2178309 })();  Otherwise, a simple fold will suffice. {{Trans|Haskell}} (Memoized fold example) (() => { 'use strict'; // fib :: Int -> Int let fib = n => range(1, n) .reduce(([a, b]) => [b, a + b], [0, 1])[0]; // GENERIC [m..n] // range :: Int -> Int -> [Int] let range = (m, n) => Array.from({ length: Math.floor(n - m) + 1 }, (_, i) => m + i); // TEST return fib(32); // --> 2178309 })();  {{Out}} 2178309  ## Joy ### Recursive DEFINE fib == [small] [] [pred dup pred] [+] binrec.  ### Iterative DEFINE fib == [1 0] dip [swap [+] unary] times popd.  ## jq jq does not (yet) have infinite-precision integer arithmetic, and currently the following algorithms only give exact answers up to fib(78). At a certain point, integers are converted to floats, but floating point precision for fib(n) fails after n = 1476: in jq, fib(1476) evaluates to 1.3069892237633987e+308 ### Recursive def nth_fib_naive(n): if (n < 2) then n else nth_fib_naive(n - 1) + nth_fib_naive(n - 2) end;  ### Tail Recursive Recent versions of jq (after July 1, 2014) include basic optimizations for tail recursion, and nth_fib is defined here to take advantage of TCO. For example, nth_fib(10000000) completes with only 380KB (that's K) of memory. However nth_fib can also be used with earlier versions of jq. def nth_fib(n): # input: [f(i-2), f(i-1), countdown] def fib: (.[0] + .[1]) as sum | .[2] as n | if (n <= 0) then sum else [ .[1], sum, n - 1 ] | fib end; [-1, 1, n] | fib;  Example:  (range(0;5), 50) | [., nth_fib(.)]  yields: [0,0] [1,1] [2,1] [3,2] [4,3] [50,12586269025]  ===Binet's Formula=== def fib_binet(n): (5|sqrt) as rt | ((1 + rt)/2) as phi | ((phi | log) * n | exp) as phin | (if 0 == (n % 2) then 1 else -1 end) as sign | ( (phin - (sign / phin) ) / rt ) + .5 | floor;  ### Generator The following is a jq generator which produces the first n terms of the Fibonacci sequence efficiently, one by one. Notice that it is simply a variant of the above tail-recursive function. The function is in effect turned into a generator by changing "( _ | fib )" to "sum, (_ | fib)". # Generator def fibonacci(n): # input: [f(i-2), f(i-1), countdown] def fib: (.[0] + .[1]) as sum | if .[2] == 0 then sum else sum, ([ .[1], sum, .[2] - 1 ] | fib) end; [-1, 1, n] | fib;  ## Julia ### Recursive fib(n) = n < 2 ? n : fib(n-1) + fib(n-2)  ### Iterative function fib(n) x,y = (0,1) for i = 1:n x,y = (y, x+y) end x end  ### Matrix form fib(n) = ([1 1 ; 1 0]^n)[1,2]  ## K ### Recursive {:[x<3;1;_f[x-1]+_f[x-2]]}  ### Recursive with memoization Using a (global) dictionary c. {c::.();{v:c[a:x];:[x<3;1;:[_n~v;c[a]:_f[x-1]+_f[x-2];v]]}x}  ### Analytic phi:(1+_sqrt(5))%2 {_((phi^x)-((1-phi)^x))%_sqrt[5]}  ### Sequence to n {(x(|+$$\1 1)[;1]}  {x{x,+/-2#x}/!2}  ## Kabap ### Sequence to n  // Calculate the$n'th Fibonacci number

// Set this to how many in the sequence to generate
$n = 10; // These are what hold the current calculation$a = 0;
$b = 1; // This holds the complete sequence that is generated$sequence = "";

// Prepare a loop
$i = 0; :calcnextnumber;$i = $i++; // Do the calculation for this loop iteration$b = $a +$b;
$a =$b - $a; // Add the result to the sequence$sequence = $sequence <<$a;

// Make the loop run a fixed number of times
if $i <$n; {
$sequence =$sequence << ", ";
goto calcnextnumber;
}

// Use the loop counter as the placeholder
$i--; // Return the sequence return = "Fibonacci number " <<$i << " is " << $a << " (" <<$sequence << ")";



## Kotlin

enum class Fibonacci {
ITERATIVE {
override fun invoke(n: Long) = if (n < 2) {
n
} else {
var n1 = 0L
var n2 = 1L
var i = n
do {
val sum = n1 + n2
n1 = n2
n2 = sum
} while (i-- > 1)
n1
}
},
RECURSIVE {
override fun invoke(n: Long): Long = if (n < 2) n else this(n - 1) + this(n - 2)
};

abstract operator fun invoke(n: Long): Long
}

fun main(a: Array<String>) {
val r = 0..30L
Fibonacci.values().forEach {
print("${it.name}: ") r.forEach { i -> print(" " + it(i)) } println() } }  {{out}} ITERATIVE: 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040 RECURSIVE: 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040  ## L++ (defn int fib (int n) (return (? (< n 2) n (+ (fib (- n 1)) (fib (- n 2)))))) (main (prn (fib 30)))  ## LabVIEW {{VI solution|LabVIEW_Fibonacci_sequence.png}} ## Lang5 [] '__A set : dip swap __A swap 2 compress collapse '__A set execute __A -1 extract nip ; : nip swap drop ; : tuck swap over ; : -rot rot rot ; : 0= 0 == ; : 1+ 1 + ; : 1- 1 - ; : sum '+ reduce ; : bi 'keep dip execute ; : keep over 'execute dip ; : fib dup 1 > if dup 1- fib swap 2 - fib + then ; : fib dup 1 > if "1- fib" "2 - fib" bi + then ;  ## Langur val .fibonacci = f if(.x < 2: .x ; self(.x - 1) + self(.x - 2)) writeln map .fibonacci, series 2..20  {{out}} [1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765]  ## lambdatalk  1) basic version {def fib1 {lambda {:n} {if {< :n 3} then 1 else {+ {fib1 {- :n 1}} {fib1 {- :n 2}}} }}} {fib1 16} -> 987 (CPU ~ 16ms) {fib1 30} = 832040 (CPU > 12000ms) 2) tail-recursive version {def fib2 {def fib2.r {lambda {:a :b :i} {if {< :i 1} then :a else {fib2.r :b {+ :a :b} {- :i 1}} }}} {lambda {:n} {fib2.r 0 1 :n}}} {fib2 16} -> 987 (CPU ~ 1ms) {fib2 30} -> 832040 (CPU ~2ms) {fib2 1000} -> 4.346655768693743e+208 (CPU ~ 22ms) 3) Dijkstra Algorithm {def fib3 {def fib3.r {lambda {:a :b :p :q :count} {if {= :count 0} then :b else {if {= {% :count 2} 0} then {fib3.r :a :b {+ {* :p :p} {* :q :q}} {+ {* :q :q} {* 2 :p :q}} {/ :count 2}} else {fib3.r {+ {* :b :q} {* :a :q} {* :a :p}} {+ {* :b :p} {* :a :q}} :p :q {- :count 1}} }}}} {lambda {:n} {fib3.r 1 0 0 1 :n} }} {fib3 16} -> 987 (CPU ~ 2ms) {fib3 30} -> 832040 (CPU ~ 2ms) {fib3 1000} -> 4.346655768693743e+208 (CPU ~ 3ms) 4) memoization {def fib4 {def fib4.m {array.new}} // init an empty array {def fib4.r {lambda {:n} {if {< :n 2} then {array.get {array.set! {fib4.m} :n 1} :n} // init with 1,1 else {if {equal? {array.get {fib4.m} :n} undefined} // if not exists then {array.get {array.set! {fib4.m} :n {+ {fib4.r {- :n 1}} {fib4.r {- :n 2}}}} :n} // compute it else {array.get {fib4.m} :n} }}}} // else get it {lambda {:n} {fib4.r :n} {fib4.m} }} // display the number AND all its predecessors -> fib4 {fib4 90} -> 4660046610375530000 [1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657,46368,75025,121393,196418, 317811,514229,832040,1346269,2178309,3524578,5702887,9227465,14930352,24157817,39088169,63245986,102334155, 165580141,267914296,433494437,701408733,1134903170,1836311903,2971215073,4807526976,7778742049,12586269025, 20365011074,32951280099,53316291173,86267571272,139583862445,225851433717,365435296162,591286729879,956722026041, 1548008755920,2504730781961,4052739537881,6557470319842,10610209857723,17167680177565,27777890035288,44945570212853, 72723460248141,117669030460994,190392490709135,308061521170129,498454011879264,806515533049393,1304969544928657, 2111485077978050,3416454622906707,5527939700884757,8944394323791464,14472334024676220,23416728348467684, 37889062373143900,61305790721611580,99194853094755490,160500643816367070,259695496911122560,420196140727489660, 679891637638612200,1100087778366101900,1779979416004714000,2880067194370816000,4660046610375530000] 5) Binet's formula (non recursive) {def fib5 {lambda {:n} {let { {:n :n} {:sqrt5 {sqrt 5}} } {round {/ {- {pow {/ {+ 1 :sqrt5} 2} :n} {pow {/ {- 1 :sqrt5} 2} :n}} :sqrt5}}} }} {fib5 16} -> 987 (CPU ~ 1ms) {fib5 30} -> 832040 (CPU ~ 1ms) {fib5 1000} -> 4.346655768693743e+208 (CPU ~ 1ms)  ## Lasso  define fibonacci(n::integer) => { #n < 1 ? return false local( swap = 0, n1 = 0, n2 = 1 ) loop(#n) => { #swap = #n1 + #n2; #n2 = #n1; #n1 = #swap; } return #n1 } fibonacci(0) //->output false fibonacci(1) //->output 1 fibonacci(2) //->output 1 fibonacci(3) //->output 2  ## Latitude ### Recursive fibo := { takes '[n]. if { n <= 1. } then { n. } else { fibo (n - 1) + fibo (n - 2). }. }.  ### Memoization fibo := { takes '[n]. cache := here cache. { cache slot? (n ordinal). } ifFalse { cache slot (n ordinal) = if { n <= 1. } then { n. } else { fibo (n - 1) + fibo (n - 2). }. }. cache slot (n ordinal). } tap { ;; Attach the cache to the method object itself. #'self cache := Object clone. }.  ## LFE ### Recursive  (defun fib ((0) 0) ((1) 1) ((n) (+ (fib (- n 1)) (fib (- n 2)))))  ### Iterative  (defun fib ((n) (when (>= n 0)) (fib n 0 1))) (defun fib ((0 result _) result) ((n result next) (fib (- n 1) next (+ result next))))  ## Liberty BASIC ### Iterative/Recursive  for i = 0 to 15 print fiboR(i),fiboI(i) next i function fiboR(n) if n <= 1 then fiboR = n else fiboR = fiboR(n-1) + fiboR(n-2) end if end function function fiboI(n) a = 0 b = 1 for i = 1 to n temp = a + b a = b b = temp next i fiboI = a end function  {{out}}  0 0 1 1 1 1 2 2 3 3 5 5 8 8 13 13 21 21 34 34 55 55 89 89 144 144 233 233 377 377 610 610  ### Iterative/Negative  print "Rosetta Code - Fibonacci sequence": print print " n Fn" for x=-12 to 12 '68 max print using("### ", x); using("##############", FibonacciTerm(x)) next x print [start] input "Enter a term#: "; n$
n$=lower$(trim$(n$))
if n$="" then print "Program complete.": end print FibonacciTerm(val(n$))
goto [start]

function FibonacciTerm(n)
n=int(n)
FTa=0: FTb=1: FTc=-1
select case
case n=0  : FibonacciTerm=0  : exit function
case n=1  : FibonacciTerm=1  : exit function
case n=-1 : FibonacciTerm=-1 : exit function
case n>1
for x=2 to n
FibonacciTerm=FTa+FTb
FTa=FTb: FTb=FibonacciTerm
next x
exit function
case n<-1
for x=-2 to n step -1
FibonacciTerm=FTa+FTc
FTa=FTc: FTc=FibonacciTerm
next x
exit function
end select
end function



{{out}}


Rosetta Code - Fibonacci sequence

n             Fn
-12           -144
-11            -89
-10            -55
-9            -34
-8            -21
-7            -13
-6             -8
-5             -5
-4             -3
-3             -2
-2             -1
-1             -1
0              0
1              1
2              1
3              2
4              3
5              5
6              8
7             13
8             21
9             34
10             55
11             89
12            144

Enter a term#: 12
144
Enter a term#:
Program complete.



## Lingo

### Recursive

on fib (n)
if n<2 then return n
return fib(n-1)+fib(n-2)
end


### Iterative

on fib (n)
if n<2 then return n
fibPrev = 0
fib = 1
repeat with i = 2 to n
tmp = fib
fib = fib + fibPrev
fibPrev = tmp
end repeat
return fib
end


### Analytic

on fib (n)
sqrt5 = sqrt(5.0)
p = (1+sqrt5)/2
q = 1 - p
return integer((power(p,n)-power(q,n))/sqrt5)
end


## Lisaac

- fib(n : UINTEGER_32) : UINTEGER_64 <- (
+ result : UINTEGER_64;
(n < 2).if {
result := n;
} else {
result := fib(n - 1) + fib(n - 2);
};
result
);


## LiveCode

-- Iterative, translation of the basic version.
function fibi n
put 0 into aa
put 1 into b
repeat with i = 1 to n
put aa + b into temp
put b into aa
put temp into b
end repeat
return aa
end fibi

-- Recursive
function fibr n
if n <= 1 then
return n
else
return fibr(n-1) + fibr(n-2)
end if
end fibr


## LLVM

; This is not strictly LLVM, as it uses the C library function "printf".
; LLVM does not provide a way to print values, so the alternative would be
; to just load the string into memory, and that would be boring.

; Additional comments have been inserted, as well as changes made from the output produced by clang such as putting more meaningful labels for the jumps

"PRINT_LONG" = comdat any @"PRINT_LONG" = linkonce_odr unnamed_addr constant [5 x i8] c"%ld\0A\00", comdat, align 1 ;--- The declaration for the external C printf function. declare i32 @printf(i8*, ...) ;-------------------------------------------------------------------- ;-- Function for calculating the nth fibonacci numbers ;-------------------------------------------------------------------- define i32 @fibonacci(i32) { %2 = alloca i32, align 4 ;-- allocate local copy of n %3 = alloca i32, align 4 ;-- allocate a %4 = alloca i32, align 4 ;-- allocate b store i32 %0, i32* %2, align 4 ;-- store copy of n store i32 0, i32* %3, align 4 ;-- a := 0 store i32 1, i32* %4, align 4 ;-- b := 1 br label %loop loop: %5 = load i32, i32* %2, align 4 ;-- load n %6 = icmp sgt i32 %5, 0 ;-- n > 0 br i1 %6, label %loop_body, label %exit loop_body: %7 = load i32, i32* %3, align 4 ;-- load a %8 = load i32, i32* %4, align 4 ;-- load b %9 = add nsw i32 %7, %8 ;-- t = a + b store i32 %8, i32* %3, align 4 ;-- store a = b store i32 %9, i32* %4, align 4 ;-- store b = t %10 = load i32, i32* %2, align 4 ;-- load n %11 = add nsw i32 %10, -1 ;-- decrement n store i32 %11, i32* %2, align 4 ;-- store n br label %loop exit: %12 = load i32, i32* %3, align 4 ;-- load a ret i32 %12 ;-- return a } ;-------------------------------------------------------------------- ;-- Main function for printing successive fibonacci numbers ;-------------------------------------------------------------------- define i32 @main() { %1 = alloca i32, align 4 ;-- allocate index store i32 0, i32* %1, align 4 ;-- index := 0 br label %loop loop: %2 = load i32, i32* %1, align 4 ;-- load index %3 = icmp sle i32 %2, 12 ;-- index <= 12 br i1 %3, label %loop_body, label %exit loop_body: %4 = load i32, i32* %1, align 4 ;-- load index %5 = call i32 @fibonacci(i32 %4) %6 = call i32 (i8*, ...) @printf(i8* getelementptr inbounds ([5 x i8], [5 x i8]* @"PRINT_LONG", i32 0, i32 0), i32 %5) %7 = load i32, i32* %1, align 4 ;-- load index %8 = add nsw i32 %7, 1 ;-- increment index store i32 %8, i32* %1, align 4 ;-- store index br label %loop exit: ret i32 0 ;-- return EXIT_SUCCESS }  {{out}} 0 1 1 2 3 5 8 13 21 34 55 89 144  to fib :n [:a 0] [:b 1] if :n < 1 [output :a] output (fib :n-1 :b :a+:b) end  ## LOLCODE  HAI 1.2 HOW DUZ I fibonacci YR N EITHER OF BOTH SAEM N AN 1 AN BOTH SAEM N AN 0 O RLY? YA RLY, FOUND YR 1 NO WAI I HAS A N1 I HAS A N2 N1 R DIFF OF N AN 1 N2 R DIFF OF N AN 2 N1 R fibonacci N1 N2 R fibonacci N2 FOUND YR SUM OF N1 AN N2 OIC IF U SAY SO KTHXBYE  ## Lua ### Recursive  --calculates the nth fibonacci number. Breaks for negative or non-integer n. function fibs(n) return n < 2 and n or fibs(n - 1) + fibs(n - 2) end  ### Pedantic Recursive  --more pedantic version, returns 0 for non-integer n function pfibs(n) if n ~= math.floor(n) then return 0 elseif n < 0 then return pfibs(n + 2) - pfibs(n + 1) elseif n < 2 then return n else return pfibs(n - 1) + pfibs(n - 2) end end  ### Tail Recursive  function a(n,u,s) if n<2 then return u+s end return a(n-1,u+s,u) end function trfib(i) return a(i-1,1,0) end  ### Table Recursive  fib_n = setmetatable({1, 1}, {__index = function(z,n) return n<=0 and 0 or z[n-1] + z[n-2] end})  ### Table Recursive 2  -- table recursive done properly (values are actually saved into table; -- also the first element of Fibonacci sequence is 0, so the initial table should be {0, 1}). fib_n = setmetatable({0, 1}, { __index = function(t,n) if n <= 0 then return 0 end t[n] = t[n-1] + t[n-2] return t[n] end })  ### Iterative  function ifibs(n) local p0,p1=0,1 for _=1,n do p0,p1 = p1,p0+p1 end return p0 end  ## Luck function fib(x: int): int = ( let cache = {} in let fibc x = if x<=1 then x else ( if x not in cache then cache[x] = fibc(x-1) + fibc(x-2); cache[x] ) in fibc(x) );; for x in range(10) do print(fib(x))  ## Lush (de fib-rec (n) (if (< n 2) n (+ (fib-rec (- n 2)) (fib-rec (- n 1)))))  ## LSL Rez a box on the ground, and add the following as a New Script. integer Fibonacci(integer n) { if(n<2) { return n; } else { return Fibonacci(n-1)+Fibonacci(n-2); } } default { state_entry() { integer x = 0; for(x=0 ; x<35 ; x++) { llOwnerSay("Fibonacci("+(string)x+")="+(string)Fibonacci(x)); } } }  Output:  Fibonacci(0)=0 Fibonacci(1)=1 Fibonacci(2)=1 Fibonacci(3)=2 Fibonacci(4)=3 Fibonacci(5)=5 Fibonacci(6)=8 Fibonacci(7)=13 Fibonacci(8)=21 Fibonacci(9)=34 Fibonacci(10)=55 Fibonacci(11)=89 Fibonacci(12)=144 Fibonacci(13)=233 Fibonacci(14)=377 Fibonacci(15)=610 Fibonacci(16)=987 Fibonacci(17)=1597 Fibonacci(18)=2584 Fibonacci(19)=4181 Fibonacci(20)=6765 Fibonacci(21)=10946 Fibonacci(22)=17711 Fibonacci(23)=28657 Fibonacci(24)=46368 Fibonacci(25)=75025 Fibonacci(26)=121393 Fibonacci(27)=196418 Fibonacci(28)=317811 Fibonacci(29)=514229 Fibonacci(30)=832040 Fibonacci(31)=1346269 Fibonacci(32)=2178309 Fibonacci(33)=3524578 Fibonacci(34)=5702887  ## M2000 Interpreter Return decimal type and use an Inventory (as closure) to store known return values. All closures are in scope in every recursive call (we use here lambda(), but we can use fib(), If we make Fib1=fib then we have to use lambda() for recursion.  Inventory K=0:=0,1:=1 fib=Lambda K (x as decimal)-> { If Exist(K, x) Then =Eval(K) :Exit Def Ret as Decimal Ret=If(x>1->Lambda(x-1)+Lambda(x-2), x) Append K, x:=Ret =Ret } \\ maximum 139 For i=1 to 139 { Print Fib(i) }  Here an example where we use a BigNum class to make a Group which hold a stack of values, and take 14 digits per item in stack. We can use inventory to hold groups, so we use the fast fib() function from code above, where we remove the type definition of Ret variable, and set two first items in inventory as groups.  Class BigNum { a=stack Function Digits { =len(.a)*14-(14-len(str(stackitem(.a,len(.a)) ,"")))
}
Operator "+" (n) {
\\ we get a copy, but .a is pointer
\\ we make a copy, and get a new pointer
.a<=stack(.a)
acc=0
carry=0
const d=100000000000000@
k=min.data(Len(.a), len(n.a))
i=each(.a, 1,k )
j=each(n.a, 1,k)
while  i, j {
acc=stackitem(i)+stackitem(j)+carry
carry= acc div d
return .a, i^+1:=acc mod d
}
if len(.a)<len(n.a) Then  {
i=each(n.a, k+1, -1)
while i {
acc=stackitem(i)+carry
carry= acc div d
stack .a  {data acc mod d}
}
} ELse.if len(.a)>len(n.a) Then  {
i=each(.a, k+1, -1)
while i {
acc=stackitem(i)+carry
carry= acc div d
Return .a, i^+1:=acc mod d
if carry else exit
}
}
if carry then stack .a { data carry}
}
Function tostring${ if len(.a)=0 then ="0" : Exit if len(.a)=1 then =str$(Stackitem(.a),"") : Exit
document buf$=str$(Stackitem(.a, len(.a)),"")
for i=len(.a)-1 to  1 {
Stack .a {
buf$=str$(StackItem(i), "00000000000000")
}
}
=buf$} class: Module BigNum (s$) {
s$=filter$(s$,"+-.,") if s$<>""  Then {
repeat {
If len(s$)<14 then Stack .a { Data val(s$) }: Exit
Stack .a { Data  val(Right$(s$, 14)) }
S$=Left$(S$, len(S$)-14)
} Until S$="" } } } Inventory K=0:=BigNum("0"),1:=BigNum("1") fib=Lambda K (x as decimal)-> { If Exist(K, x) Then =Eval(K) :Exit Ret=If(x>1->Lambda(x-1)+Lambda(x-2), bignum(str$(x,"")))
Append K, x:=Ret
=Ret
}
\\ Using this to handle form  refresh by code
Set Fast!
For i=1 to 4000 {
N=Fib(i)
Print i
loop(0,15,fibo')


## Maple


> f := n -> ifelse(n<3,1,f(n-1)+f(n-2));
> f(2);
1
> f(3);
2



=={{header|Mathematica}} / {{header|Wolfram Language}}== The Wolfram Language already has a built-in function Fibonacci, but a simple recursive implementation would be

fib[0] = 0
fib[1] = 1
fib[n_Integer] := fib[n - 1] + fib[n - 2]


An optimization is to cache the values already calculated:

fib[0] = 0
fib[1] = 1
fib[n_Integer] := fib[n] = fib[n - 1] + fib[n - 2]


The above implementations may be too simplistic, as the first is incredibly slow for any reasonable range due to nested recursions and while the second is faster it uses an increasing amount of memory. The following uses recursion much more effectively while not using memory:

fibi[prvprv_Integer, prv_Integer, rm_Integer] :=
If[rm < 1, prvprv, fibi[prv, prvprv + prv, rm - 1]]
fib[n_Integer] := fibi[0, 1, n]


However, the recursive approaches in Mathematica are limited by the limit set for recursion depth (default 1024 or 4096 for the above cases), limiting the range for 'n' to about 1000 or 2000. The following using an iterative approach has an extremely high limit (greater than a million):

fib[n_Integer] := Block[{tmp, prvprv = 0, prv = 1},
For[i = 0, i < n, i++, tmp = prv; prv += prvprv; prvprv = tmp];
Return[prvprv]]


If one wanted a list of Fibonacci numbers, the following is quite efficient:

fibi[{prvprv_Integer, prv_Integer}] := {prv, prvprv + prv}
fibList[n_Integer] := Map[Take[#, 1] &, NestList[fibi, {0, 1}, n]] // Flatten


Output from the last with "fibList[100]":

{0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, \
1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, \
196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, \
9227465, 14930352, 24157817, 39088169, 63245986, 102334155, \
165580141, 267914296, 433494437, 701408733, 1134903170, 1836311903, \
2971215073, 4807526976, 7778742049, 12586269025, 20365011074, \
32951280099, 53316291173, 86267571272, 139583862445, 225851433717, \
365435296162, 591286729879, 956722026041, 1548008755920, \
2504730781961, 4052739537881, 6557470319842, 10610209857723, \
17167680177565, 27777890035288, 44945570212853, 72723460248141, \
117669030460994, 190392490709135, 308061521170129, 498454011879264, \
806515533049393, 1304969544928657, 2111485077978050, \
3416454622906707, 5527939700884757, 8944394323791464, \
14472334024676221, 23416728348467685, 37889062373143906, \
61305790721611591, 99194853094755497, 160500643816367088, \
259695496911122585, 420196140727489673, 679891637638612258, \
1100087778366101931, 1779979416004714189, 2880067194370816120, \
4660046610375530309, 7540113804746346429, 12200160415121876738, \
19740274219868223167, 31940434634990099905, 51680708854858323072, \
83621143489848422977, 135301852344706746049, 218922995834555169026, \
354224848179261915075}


The Wolfram Language can also solve recurrence equations using the built-in function RSolve

fib[n] /. RSolve[{fib[n] == fib[n - 1] + fib[n - 2], fib[0] == 0,
fib[1] == 1}, fib[n], n][[1]]


which evaluates to the built-in function Fibonacci[n]

This function can also be expressed as

Fibonacci[n] // FunctionExpand // FullSimplify


which evaluates to

(2^-n ((1 + Sqrt[5])^n - (-1 + Sqrt[5])^n Cos[n π]))/Sqrt[5]


and is defined for all real or complex values of n.

## MATLAB

### Matrix

{{trans|Julia}}

function f = fib(n)

f = [1 1 ; 1 0]^(n-1);
f = f(1,1);

end


### Iterative

function F = fibonacci(n)

Fn = [1 0]; %Fn(1) is F_{n-2}, Fn(2) is F_{n-1}
F = 0; %F is F_{n}

for i = (1:abs(n))
Fn(2) = F;
F = sum(Fn);
Fn(1) = Fn(2);
end

if n < 0
F = F*((-1)^(n+1));
end

end


The MATLAB help file suggests an interesting method of generating the Fibonacci numbers. Apparently the determinate of the Dramadah Matrix of type 3 (MATLAB designation) and size n-by-n is the nth Fibonacci number. This method is implimented below.

function number = fibonacci2(n)

if n == 1
number = 1;
elseif n == 0
number = 0;
elseif n < 0
number = ((-1)^(n+1))*fibonacci2(-n);;
else
end

end


### Tartaglia/Pascal Triangle Method


function number = fibonacci(n)
%construct the Tartaglia/Pascal Triangle
pt=tril(ones(n));
for r = 3 : n
% Every element is the addition of the two elements
% on top of it. That means the previous row.
for c = 2 : r-1
pt(r, c) = pt(r-1, c-1) + pt(r-1, c);
end
end
number=trace(rot90(pt));
end



## Maxima

/* fib(n) is built-in; here is an implementation */
fib2(n) := (matrix([0, 1], [1, 1])^^n)[1, 2]$fib2(100)-fib(100); 0 fib2(-10); -55  ## MAXScript ### Iterative fn fibIter n = ( if n < 2 then ( n ) else ( fib = 1 fibPrev = 1 for num in 3 to n do ( temp = fib fib += fibPrev fibPrev = temp ) fib ) )  ### Recursive fn fibRec n = ( if n < 2 then ( n ) else ( fibRec (n - 1) + fibRec (n - 2) ) )  ## Mercury Mercury is both a logic language and a functional language. As such there are two possible interfaces for calculating a Fibonacci number. This code shows both styles. Note that much of the code here is ceremony put in place to have this be something which can actually compile. The actual Fibonacci number generation is contained in the predicate fib/2 and in the function fib/1. The predicate main/2 illustrates first the unification semantics of the predicate form and the function call semantics of the function form. The provided code uses a very naive form of generating a Fibonacci number. A more realistic implementation would use memoization to cache previous results, exchanging time for space. Also, in the case of supplying both a function implementation and a predicate implementation, one of the two would be implemented in terms of the other. Examples of this are given as comments below. ### fib.m  % The following code is derived from the Mercury Tutorial by Ralph Becket. % http://www.mercury.csse.unimelb.edu.au/information/papers/book.pdf :- module fib. :- interface. :- import_module io. :- pred main(io::di, io::uo) is det. :- implementation. :- import_module int. :- pred fib(int::in, int::out) is det. fib(N, X) :- ( if N =< 2 then X = 1 else fib(N - 1, A), fib(N - 2, B), X = A + B ). :- func fib(int) = int is det. fib(N) = X :- fib(N, X). main(!IO) :- fib(40, X), write_string("fib(40, ", !IO), write_int(X, !IO), write_string(")\n", !IO), write_string("fib(40) = ", !IO), write_int(fib(40), !IO), write_string("\n", !IO).  ### Iterative algorithm The much faster iterative algorithm can be written as:  :- pred fib_acc(int::in, int::in, int::in, int::in, int::out) is det. fib_acc(N, Limit, Prev2, Prev1, Res) :- ( N < Limit -> % limit not reached, continue computation. ( N =< 2 -> Res0 = 1 ; Res0 = Prev2 + Prev1 ), fib_acc(N+1, Limit, Prev1, Res0, Res) ; % Limit reached, return the sum of the two previous results. Res = Prev2 + Prev1 ).  This predicate can be called as fib_acc(1, 40, 1, 1, Result)  It has several inputs which form the loop, the first is the current number, the second is a limit, ie when to stop counting. And the next two are accumulators for the last and next-to-last results. ### Memoization But what if you want the speed of the fib_acc with the recursive (more declarative) definition of fib? Then use memoization, because Mercury is a pure language fib(N, F) will always give the same F for the same N, guaranteed. Therefore memoization asks the compiler to use a table to remember the value for F for any N, and it's a one line change:  :- pragma memo(fib/2). :- pred fib(int::in, int::out) is det. fib(N, X) :- ( if N =< 2 then X = 1 else fib(N - 1, A), fib(N - 2, B), X = A + B ).  We've shown the definition of fib/2 again, but the only change here is the memoization pragma (see the reference manual). This is not part of the language specification and different Mercury implementations are allowed to ignore it, however there is only one implementation so in practice memoization is fully supported. Memoization trades speed for space, a table of results is constructed and kept in memory. So this version of fib consumes more memory than than fib_acc. It is also slightly slower than fib_acc since it must manage its table of results but it is much much faster than without memoization. Memoization works very well for the Fibonacci sequence because in the naive version the same results are calculated over and over again. ## Metafont vardef fibo(expr n) = if n=0: 0 elseif n=1: 1 else: fibo(n-1) + fibo(n-2) fi enddef; for i=0 upto 10: show fibo(i); endfor end  ## Microsoft Small Basic ### Iterative ' Fibonacci sequence - 31/07/2018 n = 139 f1 = 0 f2 = 1 TextWindow.WriteLine("fibo(0)="+f1) TextWindow.WriteLine("fibo(1)="+f2) For i = 2 To n f3 = f1 + f2 TextWindow.WriteLine("fibo("+i+")="+f3) f1 = f2 f2 = f3 EndFor  {{out}}  fibo(139)=50095301248058391139327916261  ===Binet's Formula=== ' Fibonacci sequence - Binet's Formula - 31/07/2018 n = 69 sq5=Math.SquareRoot(5) phi1=(1+sq5)/2 phi2=(1-sq5)/2 phi1n=phi1 phi2n=phi2 For i = 2 To n phi1n=phi1n*phi1 phi2n=phi2n*phi2 TextWindow.Write(Math.Floor((phi1n-phi2n)/sq5)+" ") EndFor  {{out}}  1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040 1346269 2178309 3524578 5702887 9227465 14930352 24157817 39088169 63245986 102334155 165580141 267914296 433494437 701408733 1134903170 1836311903 2971215073 4807526976 7778742049 12586269025 20365011074 32951280099 53316291173 86267571272 139583862445 225851433717 365435296162 591286729879 956722026041 1548008755920 2504730781961 4052739537881 6557470319842 10610209857723 17167680177565 27777890035288 44945570212853 72723460248141 117669030460994  ## min {{works with|min|0.19.3}} ( (2 <) ((0 1 (dup rollup +)) dip pred times nip) unless ) :fib  ## MiniScript fibonacci = function(n) if n < 2 then return n ans = 0 n1 = 0 n2 = 1 for i in range(n-1, 1) ans = n1 + n2 n1 = n2 n2 = ans end for return ans end function print fibonacci(6)  ## Mirah def fibonacci(n:int) return n if n < 2 fibPrev = 1 fib = 1 3.upto(Math.abs(n)) do oldFib = fib fib = fib + fibPrev fibPrev = oldFib end fib * (n<0 ? int(Math.pow(n+1, -1)) : 1) end puts fibonacci 1 puts fibonacci 2 puts fibonacci 3 puts fibonacci 4 puts fibonacci 5 puts fibonacci 6 puts fibonacci 7  ## MIPS Assembly This is the iterative approach to the Fibonacci sequence.  .text main: li$v0, 5		# read integer from input. The read integer will be stroed in $v0 syscall beq$v0, 0, is1
beq	$v0, 1, is1 li$s4, 1		# the counter which has to equal to $v0 li$s0, 1
li	$s1, 1 loop: add$s2, $s0,$s1
addi	$s4,$s4, 1
beq	$v0,$s4, iss2

add	$s0,$s1, $s2 addi$s4, $s4, 1 beq$v0, $s4, iss0 add$s1, $s2,$s0
addi	$s4,$s4, 1
beq	$v0,$s4, iss1

b 	loop

iss0:	move	$a0,$s0
b	print

iss1:	move	$a0,$s1
b	print

iss2:	move	$a0,$s2
b	print

is1:	li	$a0, 1 b print print: li$v0, 1
syscall
li	$v0, 10 syscall  =={{header|МК-61/52}}== П0 1 lg Вx <-> + L0 03 С/П БП 03  Instruction: ''n'' В/О С/П, where ''n'' is serial number of the number of Fibonacci sequence; С/П for the following numbers. ## ML = ## Standard ML = ### =Recursion= This version is tail recursive. sml fun fib n = let fun fib' (0,a,b) = a | fib' (n,a,b) = fib' (n-1,a+b,a) in fib' (n,0,1) end  = ## MLite = ### =Recursion= Tail recursive. fun fib (0, x1, x2) = x2 | (n, x1, x2) = fib (n-1, x2, x1+x2) | n = fib (n, 0, 1)  ## ML/I MCSKIP "WITH" NL "" Fibonacci - recursive MCSKIP MT,<> MCINS %. MCDEF FIB WITHS () AS <MCSET T1=%A1. MCGO L1 UNLESS 2 GR T1 %T1.<>MCGO L0 %L1.%FIB(%T1.-1)+FIB(%T1.-2).> fib(0) is FIB(0) fib(1) is FIB(1) fib(2) is FIB(2) fib(3) is FIB(3) fib(4) is FIB(4) fib(5) is FIB(5)  =={{header|Modula-2}}== MODULE Fibonacci; FROM FormatString IMPORT FormatString; FROM Terminal IMPORT WriteString,WriteLn,ReadChar; PROCEDURE Fibonacci(n : LONGINT) : LONGINT; VAR a,b,c : LONGINT; BEGIN IF n<0 THEN RETURN 0 END; a:=1; b:=1; WHILE n>0 DO c := a + b; a := b; b := c; DEC(n) END; RETURN a END Fibonacci; VAR buf : ARRAY[0..63] OF CHAR; i : INTEGER; r : LONGINT; BEGIN FOR i:=0 TO 10 DO r := Fibonacci(i); FormatString("%l\n", buf, r); WriteString(buf); END; ReadChar END Fibonacci.  =={{header|Modula-3}}== ### Recursive PROCEDURE Fib(n: INTEGER): INTEGER = BEGIN IF n < 2 THEN RETURN n; ELSE RETURN Fib(n-1) + Fib(n-2); END; END Fib;  === Iterative (with negatives) === PROCEDURE IterFib(n: INTEGER): INTEGER = VAR limit := ABS(n); prev := 0; curr, next: INTEGER; BEGIN (* trivial case *) IF n = 0 THEN RETURN 0; END; IF n > 0 THEN (* positive case *) curr := 1; FOR i := 2 TO limit DO next := prev + curr; prev := curr; curr := next; END; ELSE (* negative case *) curr := -1; FOR i := 2 TO limit DO next := prev - curr; prev := curr; curr := next; END; END; RETURN curr; END IterFib;  ## Monicelli Recursive version. It includes a main that reads a number N from standard input and prints the Nth Fibonacci number.  # Main Lei ha clacsonato voglio un nonnulla, Necchi mi porga un nonnulla il nonnulla come se fosse brematurata la supercazzola bonaccia con il nonnulla o scherziamo? un nonnulla a posterdati # Fibonacci function 'bonaccia' blinda la supercazzola Necchi bonaccia con antani Necchi o scherziamo? che cos'è l'antani? minore di 3: vaffanzum 1! o tarapia tapioco: voglio unchiamo, Necchi come se fosse brematurata la supercazzola bonaccia con antani meno 1 o scherziamo? voglio duechiamo, Necchi come se fosse brematurata la supercazzola bonaccia con antani meno 2 o scherziamo? vaffanzum unchiamo più duechiamo! e velocità di esecuzione  ## MontiLang Reads number from standard input and prints to that number in the fibonacci sequence 0 VAR a . 1 VAR b . INPUT TOINT FOR : a b + VAR c . a PRINT . b VAR a . c VAR b . ENDFOR  Forth-style solution def over swap dup rot swap enddef |Enter a number to obtain Fibonacci sequence: | input nip var count . 0 1 FOR count over out |, | out . + swap ENDFOR . print input clear  Simpler |Enter a number to obtain Fibonacci sequence: | input nip 1 - var count . 0 1 FOR count out |, | out . dup rot + ENDFOR print input /# wait until press ENTER #/ clear /# empties the stack #/  ## MUMPS ### Iterative FIBOITER(N) ;Iterative version to get the Nth Fibonacci number ;N must be a positive integer ;F is the tree containing the values ;I is a loop variable. QUIT:(N\1'=N)!(N<0) "Error: "_N_" is not a positive integer." NEW F,I SET F(0)=0,F(1)=1 QUIT:N<2 F(N) FOR I=2:1:N SET F(I)=F(I-1)+F(I-2) QUIT F(N)   USER>W $$FIBOITER^ROSETTA(30) 832040  ## Nanoquery ### Iterative def fibIter(n) if (n < 2) return n end if fib = 1 fibPrev = 1 for (num = 2) (num < n) (num = num+1) fib = (fib + fibPrev) fibPrev = (fib - fibPrev) end for return fib end fibIter  ## Nemerle ### Recursive using System; using System.Console; module Fibonacci { Fibonacci(x : long) : long { |x when x < 2 => x |_ => Fibonacci(x - 1) + Fibonacci(x - 2) } Main() : void { def num = Int64.Parse(ReadLine()); foreach (n in [0 .. num]) WriteLine("{0}: {1}", n, Fibonacci(n)); } }  ### Tail Recursive Fibonacci(x : long, current : long, next : long) : long { match(x) { |0 => current |_ => Fibonacci(x - 1, next, current + next) } } Fibonacci(x : long) : long { Fibonacci(x, 0, 1) }  ## NESL ### Recursive function fib(n) = if n < 2 then n else fib(n - 2) + fib(n - 1);  ## NetRexx {{trans|REXX}} /* NetRexx */ options replace format comments java crossref savelog symbols numeric digits 210000 /*prepare for some big 'uns. */ parse arg x y . /*allow a single number or range.*/ if x == '' then do /*no input? Then assume -30-->+30*/ x = -30 y = -x end if y == '' then y = x /*if only one number, show fib(n)*/ loop k = x to y /*process each Fibonacci request.*/ q = fib(k) w = q.length /*if wider than 25 bytes, tell it*/ say 'Fibonacci' k"="q if w > 25 then say 'Fibonacci' k "has a length of" w end k exit /*-------------------------------------FIB subroutine (non-recursive)---*/ method fib(arg) private static parse arg n na = n.abs if na < 2 then return na /*handle special cases. */ a = 0 b = 1 loop j = 2 to na s = a + b a = b b = s end j if n > 0 | na // 2 == 1 then return s /*if positive or odd negative... */ else return -s /*return a negative Fib number. */  ## NewLISP ### Iterative (define (fibonacci n) (let (L '(0 1)) (dotimes (i n) (setq L (list (L 1) (apply + L)))) (L 1)) )  ### Recursive (define (fibonacci n) (if (< n 2) 1 (+ (fibonacci (- n 1)) (fibonacci (- n 2)))))  ### Matrix multiplication (define (fibonacci n) (letn (f '((0 1) (1 1)) fib f) (dotimes (i n) (set 'fib (multiply fib f))) (fib 0 1)) ) (print(fibonacci 10)) ;;89  ## NGS ### Iterative {{trans|Python}} F fib(n:Int) { n < 2 returns n local a = 1, b = 1 # i is automatically local because of for() for(i=2; i<n; i=i+1) { local next = a + b a = b b = next } b }  ## Nial ### Iterative On my machine, about 1.7s for 100,000 iterations, n=92. Maybe a few percent faster than iterative Python. Note that n>92 produces overflow; Python keeps going - single iteration with n=1,000,000 takes it about 15s. fibi is op n { if n<2 then n else x1:=0; x2:=1; for i with tell (n - 1) do x:=x1+x2; x1:=x2; x2:=x; endfor; x2 endif};  Iterative using fold. Slightly faster, <1.6s: fibf is op n {1 pick ((n- 1) fold [1 pick, +] 0 1)};  Tacit verion of above. Slightly faster still, <1.4s: fibf2 is 1 pick fold [1 pick, +] reverse (0 1 hitch) (-1+);  ### Recursive Really slow (over 8s for single iteration, n=33). (Similar to time for recursive python version with n=37.) fibr is op n {fork [2>, +, + [fibr (-1 +), fibr (-2 +)]] n};  ...or tacit version. More than twice as fast (?) but still slow: , +, + [fibr2 (-1 +), fibr2 (-2 +)]];  ### Matrix Matrix inner product (ip). This appears to be the fastest, about 1.0s for 100,000 iterations, n=92: Note that n>92 produces negative result. fibm is op n {floor (0 1 pick (reduce ip (n reshape [2 2 reshape 1 1 1 0])))};  Could it look a little more like J? (Maybe 5% slower than above.)  is reshape; ~ is tr f op a b {b f a}; % Goes before verb, rather than after like in J; _ is floor; % Not really J, but J-ish? (Cannot redefine "<.".); fibm2 is _(0 1 pick reduce ip([2 21 1 1 0](~)));  Alternate, not involving replicating matrix n times, but maybe 50% slower than the fastest matrix version above - similar speed to iterative: fibm3 is op n {a:=2 21 1 1 0; _(0 1 pick ((n- 1) fold (a ip) a))};  ## Nim ### Analytic proc Fibonacci(n: int): int64 = var fn = float64(n) var p: float64 = (1.0 + sqrt(5.0)) / 2.0 var q: float64 = 1.0 / p return int64((pow(p, fn) + pow(q, fn)) / sqrt(5.0))  ### Iterative proc Fibonacci(n: int): int = var first = 0 second = 1 for i in 0 .. <n: swap first, second second += first result = first  ### Recursive proc Fibonacci(n: int): int64 = if n <= 2: result = 1 else: result = Fibonacci(n - 1) + Fibonacci(n - 2)  ===Tail-recursive=== proc Fibonacci(n: int, current: int64, next: int64): int64 = if n == 0: result = current else: result = Fibonacci(n - 1, next, current + next) proc Fibonacci(n: int): int64 = result = Fibonacci(n, 0, 1)  ### Continuations iterator fib: int {.closure.} = var a = 0 var b = 1 while true: yield a swap a, b b = a + b var f = fib for i in 0.. <10: echo f()  =={{header|Oberon-2}}== {{Works with|oo2c Version 2}}  MODULE Fibonacci; IMPORT Out := NPCT:Console; PROCEDURE Fibs(VAR r: ARRAY OF LONGREAL); VAR i: LONGINT; BEGIN r[0] := 1.0; r[1] := 1.0; FOR i := 2 TO LEN(r) - 1 DO r[i] := r[i - 2] + r[i - 1]; END END Fibs; PROCEDURE FibsR(n: LONGREAL): LONGREAL; BEGIN IF n < 2. THEN RETURN n ELSE RETURN FibsR(n - 1) + FibsR(n - 2) END END FibsR; PROCEDURE Show(r: ARRAY OF LONGREAL); VAR i: LONGINT; BEGIN Out.String("First ");Out.Int(LEN(r),0);Out.String(" Fibonacci numbers");Out.Ln; FOR i := 0 TO LEN(r) - 1 DO Out.LongRealFix(r[i],8,0) END; Out.Ln END Show; PROCEDURE Gen(s: LONGINT); VAR x: POINTER TO ARRAY OF LONGREAL; BEGIN NEW(x,s); Fibs(x^); Show(x^) END Gen; PROCEDURE GenR(s: LONGINT); VAR i: LONGINT; BEGIN Out.String("First ");Out.Int(s,0);Out.String(" Fibonacci numbers (Recursive)");Out.Ln; FOR i := 1 TO s DO Out.LongRealFix(FibsR(i),8,0) END; Out.Ln END GenR; BEGIN Gen(10); Gen(20); GenR(10); GenR(20); END Fibonacci.  {{out}}  First 10 Fibonacci numbers 1. 1. 2. 3. 5. 8. 13. 21. 34. 55. First 20 Fibonacci numbers 1. 1. 2. 3. 5. 8. 13. 21. 34. 55. 89. 144. 233. 377. 610. 987. 1597. 2584. 4181. 6765. First 10 Fibonacci numbers (Recursive) 1. 1. 2. 3. 5. 8. 13. 21. 34. 55. First 20 Fibonacci numbers (Recursive) 1. 1. 2. 3. 5. 8. 13. 21. 34. 55. 89. 144. 233. 377. 610. 987. 1597. 2584. 4181. 6765.  ## Objeck ### Recursive bundle Default { class Fib { function : Main(args : String[]), Nil { for(i := 0; i <= 10; i += 1;) { Fib(i)->PrintLine(); }; } function : native : Fib(n : Int), Int { if(n < 2) { return n; }; return Fib(n-1) + Fib(n-2); } } }  =={{header|Objective-C}}== ### Recursive -(long)fibonacci:(int)position { long result = 0; if (position < 2) { result = position; } else { result = [self fibonacci:(position -1)] + [self fibonacci:(position -2)]; } return result; }  ### Iterative +(long)fibonacci:(int)index { long beforeLast = 0, last = 1; while (index > 0) { last += beforeLast; beforeLast = last - beforeLast; --index; } return last; }  ## OCaml ### Iterative let fib_iter n = if n < 2 then n else let fib_prev = ref 1 and fib = ref 1 in for num = 2 to n - 1 do let temp = !fib in fib := !fib + !fib_prev; fib_prev := temp done; !fib  ### Recursive let rec fib_rec n = if n < 2 then n else fib_rec (n - 1) + fib_rec (n - 2) let rec fib = function 0 -> 0 | 1 -> 1 | n -> if n > 0 then fib (n-1) + fib (n-2) else fib (n+2) - fib (n+1)  ### Arbitrary Precision Using OCaml's [http://caml.inria.fr/pub/docs/manual-ocaml/libref/Num.html Num] module. open Num let fib = let rec fib_aux f0 f1 = function | 0 -> f0 | 1 -> f1 | n -> fib_aux f1 (f1 +/ f0) (n - 1) in fib_aux (num_of_int 0) (num_of_int 1) (* support for negatives *) let fib n = if n < 0 && n mod 2 = 0 then minus_num (fib (abs n)) else fib (abs n) ;; (* It can be called from the command line with an argument *) (* Result is send to standart output *) let n = int_of_string Sys.argv.(1) in print_endline (string_of_num (fib n))  compile with: ocamlopt nums.cmxa -o fib fib.ml Output:  ./fib 0 0 ./fib 10 55 ./fib 399 108788617463475645289761992289049744844995705477812699099751202749393926359816304226 ./fib -6 -8  === O(log(n)) with arbitrary precision === This performs log2(N) matrix multiplys. Each multiplication is not constant-time but increases sub-linearly, about O(log(N)). open Num let mul (a,b,c) (d,e,f) = let bxe = b*/e in (a*/d +/ bxe, a*/e +/ b*/f, bxe +/ c*/f) let id = (Int 1, Int 0, Int 1) let rec pow a n = if n=0 then id else let b = pow a (n/2) in if (n mod 2) = 0 then mul b b else mul a (mul b b) let fib n = let (_,y,_) = (pow (Int 1, Int 1, Int 0) n) in string_of_num y ;; Printf.printf "fib %d = %s\n" 300 (fib 300)  Output: fib 300 = 222232244629420445529739893461909967206666939096499764990979600  ## Octave '''Recursive''' % recursive function fibo = recfibo(n) if ( n < 2 ) fibo = n; else fibo = recfibo(n-1) + recfibo(n-2); endif endfunction  '''Iterative''' % iterative function fibo = iterfibo(n) if ( n < 2 ) fibo = n; else f = zeros(2,1); f(1) = 0; f(2) = 1; for i = 2 : n t = f(2); f(2) = f(1) + f(2); f(1) = t; endfor fibo = f(2); endif endfunction  '''Testing''' % testing for i = 0 : 20 printf("%d %d\n", iterfibo(i), recfibo(i)); endfor  ## Oforth : fib 0 1 rot #[ tuck + ] times drop ;  ## OPL FIBON: REM Fibonacci sequence is generated to the Organiser II floating point variable limit. REM CLEAR/ON key quits. REM Mikesan - http://forum.psion2.org/ LOCAL A,B,C A=1 :B=1 :C=1 PRINT A, DO C=A+B A=B B=C PRINT A, UNTIL GET=1  ## Order ### Recursive #include <order/interpreter.h> #define ORDER_PP_DEF_8fib_rec \ ORDER_PP_FN(8fn(8N, \ 8if(8less(8N, 2), \ 8N, \ 8add(8fib_rec(8sub(8N, 1)), \ 8fib_rec(8sub(8N, 2)))))) ORDER_PP(8fib_rec(10))  Tail recursive version (example supplied with language): #include <order/interpreter.h> #define ORDER_PP_DEF_8fib \ ORDER_PP_FN(8fn(8N, \ 8fib_iter(8N, 0, 1))) #define ORDER_PP_DEF_8fib_iter \ ORDER_PP_FN(8fn(8N, 8I, 8J, \ 8if(8is_0(8N), \ 8I, \ 8fib_iter(8dec(8N), 8J, 8add(8I, 8J))))) ORDER_PP(8to_lit(8fib(8nat(5,0,0))))  ### Memoization #include <order/interpreter.h> #define ORDER_PP_DEF_8fib_memo \ ORDER_PP_FN(8fn(8N, \ 8tuple_at(0, 8fib_memo_inner(8N, 8seq)))) #define ORDER_PP_DEF_8fib_memo_inner \ ORDER_PP_FN(8fn(8N, 8M, \ 8cond((8less(8N, 8seq_size(8M)), 8pair(8seq_at(8N, 8M), 8M)) \ (8equal(8N, 0), 8pair(0, 8seq(0))) \ (8equal(8N, 1), 8pair(1, 8seq(0, 1))) \ (8else, \ 8lets((8S, 8fib_memo_inner(8sub(8N, 2), 8M)) \ (8T, 8fib_memo_inner(8dec(8N), 8tuple_at(1, 8S))) \ (8U, 8add(8tuple_at(0, 8S), 8tuple_at(0, 8T))), \ 8pair(8U, \ 8seq_append(8tuple_at(1, 8T), 8seq(8U)))))))) ORDER_PP( 8for_each_in_range(8fn(8N, 8print(8to_lit(8fib_memo(8N)) (,) 8space)), 1, 21) )  ## Oz ### Iterative Using mutable references (cells). fun{FibI N} Temp = {NewCell 0} A = {NewCell 0} B = {NewCell 1} in for I in 1..N do Temp := @A + @B A := @B B := @Temp end @A end  ### Recursive Inefficient (blows up the stack). fun{FibR N} if N < 2 then N else {FibR N-1} + {FibR N-2} end end  ===Tail-recursive=== Using accumulators. fun{Fib N} fun{Loop N A B} if N == 0 then B else {Loop N-1 A+B A} end end in {Loop N 1 0} end  ===Lazy-recursive=== declare fun lazy {FiboSeq} {LazyMap {Iterate fun { [A B]} [B A+B] end [0 1]} Head} end fun {Head A|_} A end fun lazy {Iterate F I} I|{Iterate F {F I}} end fun lazy {LazyMap Xs F} case Xs of X|Xr then {F X}|{LazyMap Xr F} [] nil then nil end end in {Show {List.take {FiboSeq} 8}}  ## PARI/GP ===Built-in=== fibonocci(n)  ### Matrix fib(n)=([1,1;1,0]^n)[1,2]  ### Analytic This uses the Binet form. fib(n)=my(phi=(1+sqrt(5))/2);round((phi^n-phi^-n)/sqrt(5))  The second term can be dropped since the error is always small enough to be subsumed by the rounding. fib(n)=round(((1+sqrt(5))/2)^n/sqrt(5))  ### Algebraic This is an exact version of the above formula. quadgen(5) represents $\phi$ and the number is stored in the form $a+b\phi$. imag takes the coefficient of $\phi$. This uses the relation :$\phi^n=F_\left\{n-1\right\}+F_n\phi$ and hence real(quadgen(5)^n) would give the (n-1)-th Fibonacci number. fib(n)=imag(quadgen(5)^n)  A more direct translation (note that $\sqrt5=2\phi-1$) would be fib(n)=my(phi=quadgen(5));(phi^n-(-1/phi)^n)/(2*phi-1)  ### Combinatorial This uses the generating function. It can be trivially adapted to give the first n Fibonacci numbers. fib(n)=polcoeff(x/(1-x-x^2)+O(x^(n+1)),n)  ### Binary powering fib(n)={ if(n<=0, if(n,(-1)^(n+1)*fib(n),0) , my(v=lucas(n-1)); (2*v[1]+v[2])/5 ) }; lucas(n)={ if (!n, return([2,1])); my(v=lucas(n >> 1), z=v[1], t=v[2], pr=v[1]*v[2]); n=n%4; if(n%2, if(n==3,[v[1]*v[2]+1,v[2]^2-2],[v[1]*v[2]-1,v[2]^2+2]) , if(n,[v[1]^2+2,v[1]*v[2]+1],[v[1]^2-2,v[1]*v[2]-1]) ) };  ### Recursive fib(n)={ if(n<2, n , fib(n-1)+fib(n) ) };  ### Anonymous recursion {{works with|PARI/GP|2.8.0+}} This uses self() which gives a self-reference. fib(n)={ if(n<2, n , my(s=self()); s(n-2)+s(n-1) ) };  It can be used without being named: apply(n->if(n<2,n,my(s=self());s(n-2)+s(n-1)), [1..10])  gives {{out}} %1 = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55]  ### Memoization F=[]; fib(n)={ if(n>#F, F=concat(F, vector(n-#F)); F[n]=fib(n-1)+fib(n-2) , if(n<2, n , if(F[n],F[n],F[n]=fib(n-1)+fib(n-2)) ) ); }  ### Iterative fib(n)={ if(n<0,return((-1)^(n+1)*fib(n))); my(a=0,b=1,t); while(n, t=a+b; a=b; b=t; n-- ); a };  ### Chebyshev This solution uses Chebyshev polynomials of the second kind (Chyebyshev U-polynomials). fib(n)=n--;polchebyshev(n,2,I/2)*I^n;  or fib(n)=abs(polchebyshev(n-1,2,I/2));  ===Anti-Hadamard matrix=== All n×n [https://en.wikipedia.org/wiki/Logical_matrix (0,1)] lower [https://en.wikipedia.org/wiki/Hessenberg_matrix Hessenberg matrices] have determinant at most F(n). The n×n anti-Hadamard matrix R. L. Graham and N. J. A. Sloane, [http://www.math.ucsd.edu/~ronspubs/84_03_anti_hadamard.pdf Anti-Hadamard matrices], Linear Algebra Appl. 62 (1984), 113–137. matches this upper bound, and hence can be used as an inefficient method for computing Fibonacci numbers of positive index. These matrices are the same as Matlab's type-3 "Dramadah" matrices, following a naming suggestion of C. L. Mallows according to Graham & Sloane. matantihadamard(n)={ matrix(n,n,i,j, my(t=j-i+1); if(t<1,t%2,t<3) ); } fib(n)=matdet(matantihadamard(n))  ### Testing adjacent bits The Fibonacci numbers can be characterized (for n > 0) as the number of n-bit strings starting and ending with 1 without adjacent 0s. This inefficient, exponential-time algorithm demonstrates: fib(n)= { my(g=2^(n+1)-1); sum(i=2^(n-1),2^n-1, bitor(i,i<<1)==g ); }  ===One-by-one=== This code is purely for amusement and requires n > 1. It tests numbers in order to see if they are Fibonacci numbers, and waits until it has seen ''n'' of them. fib(n)=my(k=0);while(n--,k++;while(!issquare(5*k^2+4)&&!issquare(5*k^2-4),k++));k  ## Pascal ### Analytic function fib(n: integer):longInt; const Sqrt5 = sqrt(5.0); C1 = ln((Sqrt5+1.0)*0.5);//ln( 1.618..) //C2 = ln((1.0-Sqrt5)*0.5);//ln(-0.618 )) tsetsetse C2 = ln((Sqrt5-1.0)*0.5);//ln(+0.618 )) begin IF n>0 then begin IF odd(n) then fib := round((exp(C1*n) + exp(C2*n) )/Sqrt5) else fib := round((exp(C1*n) - exp(C2*n) )/Sqrt5) end else Fibdirekt := 0 end;  ### Recursive function fib(n: integer): integer; begin if (n = 0) or (n = 1) then fib := n else fib := fib(n-1) + fib(n-2) end;  ### Iterative function fib(n: integer): integer; var f0, f1, tmpf0, k: integer; begin f1 := n; IF f1 >1 then begin k := f1-1; f0 := 0; f1 := 1; repeat tmpf0 := f0; f0 := f1; f1 := f1+tmpf0; dec(k); until k = 0; end else IF f1 < 0 then f1 := 0; fib := f1; end;  ### Analytic2 function FiboMax(n: integer):Extended; //maXbox begin result:= (pow((1+SQRT5)/2,n)-pow((1-SQRT5)/2,n))/SQRT5 end;  function Fibo_BigInt(n: integer): string; //maXbox var tbig1, tbig2, tbig3: TInteger; begin result:= '0' tbig1:= TInteger.create(1); //temp tbig2:= TInteger.create(0); //result (a) tbig3:= Tinteger.create(1); //b for it:= 1 to n do begin tbig1.assign(tbig2) tbig2.assign(tbig3); tbig1.add(tbig3); tbig3.assign(tbig1); end; result:= tbig2.toString(false) tbig3.free; tbig2.free; tbig1.free; end;  writeln(floattoStr(FiboMax(555))) >>>4.3516638122555E115 writeln(Fibo_BigInt(555)) >>>43516638122555047989641805373140394725407202037260729735885664398655775748034950972577909265605502785297675867877570 ## Perl ### Iterative sub fib_iter { my n = shift; use bigint try => "GMP,Pari"; my (v2,v1) = (-1,1); (v2,v1) = (v1,v2+v1) for 0..n; v1; }  ### Recursive sub fibRec { my n = shift; n < 2 ? n : fibRec(n - 1) + fibRec(n - 2); }  ### Modules Quite a few modules have ways to do this. Performance is not typically an issue with any of these until 100k or so. With GMP available, the first three are ''much'' faster at large values. # Uses GMP method so very fast use Math::AnyNum qw/fibonacci/; say fibonacci(10000); # Uses GMP method, so also very fast use Math::GMP; say Math::GMP::fibonacci(10000); # Binary ladder, GMP if available, Pure Perl otherwise use ntheory qw/lucasu/; say lucasu(1, -1, 10000); # All Perl use Math::NumSeq::Fibonacci; my seq = Math::NumSeq::Fibonacci->new; say seq->ith(10000); # All Perl use Math::Big qw/fibonacci/; say 0+fibonacci(10000); # Force scalar context # Perl, gives floating point *approximation* use Math::Fibonacci qw/term/; say term(10000);  ## Perl 6 ### List Generator This constructs the fibonacci sequence as a lazy infinite list. constant @fib = 0, 1, *+* ... *;  If you really need a function for it: sub fib (n) { @fib[n] }  To support negative indices: constant @neg-fib = 0, 1, *-* ... *; sub fib (n) { n >= 0 ?? @fib[n] !! @neg-fib[-n] }  ### Iterative sub fib (Int n --> Int) { n > 1 or return n; my (prev, this) = 0, 1; (prev, this) = this, this + prev for 1 ..^ n; return this; }  ### Recursive proto fib (Int n --> Int) {*} multi fib (0) { 0 } multi fib (1) { 1 } multi fib (n) { fib(n - 1) + fib(n - 2) }  ### Analytic sub fib (Int n --> Int) { constant φ1 = 1 / constant φ = (1 + sqrt 5)/2; constant invsqrt5 = 1 / sqrt 5; floor invsqrt5 * (φ**n + φ1**n); }  ## Phix function fibonacci(integer n) -- iterative, works for -ve numbers atom a=0, b=1 if n=0 then return 0 end if if abs(n)>=79 then ?9/0 end if -- inaccuracies creep in above 78 for i=1 to abs(n)-1 do {a,b} = {b,a+b} end for if n<0 and remainder(n,2)=0 then return -fcache[absn] end if return fcache[absn] end function for i=0 to 28 do if i then puts(1,", ") end if printf(1,"%d", fibonacci(i)) end for puts(1,"\n")  {{out}}  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811  Using native integers/atoms, errors creep in above 78, so the same program converted to use mpfr: {{libheader|mpfr}} -- demo\rosetta\fibonacci.exw include mpfr.e mpz res = NULL, prev, next integer lastn atom t0 = time() function fibonampz(integer n) -- resumable, works for -ve numbers, yields mpz integer absn = abs(n) if res=NULL or absn!=abs(lastn)+1 then if res=NULL then prev = mpz_init(0) res = mpz_init(1) next = mpz_init() else if n==lastn then return res end if end if mpz_fib2_ui(res,prev,absn) else if lastn<0 and remainder(lastn,2)=0 then mpz_mul_si(res,res,-1) end if mpz_add(next,res,prev) {prev,res,next} = {res,next,prev} end if if n<0 and remainder(n,2)=0 then mpz_mul_si(res,res,-1) end if lastn = n return res end function for i=0 to 28 do if i then puts(1,", ") end if printf(1,"%s", {mpz_get_str(fibonampz(i))}) end for puts(1,"\n") printf(1,"%s\n", {mpz_get_str(fibonampz(705))}) string s = mpz_get_str(fibonampz(4784969)) integer l = length(s) s[40..-40] = "..." ?{l,s} ?elapsed(time()-t0)  {{out}}  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811 970066202977562212558683426760773016559904631977220423547980211057068777324159443678590358026859129109599109446646966713225742014317926940054191330 {1000000,"107273956418004772293648135962250043219...407167474856539211500699706378405156269"} "2.1s"  ## PHP ### Iterative function fibIter(n) { if (n < 2) { return n; } fibPrev = 0; fib = 1; foreach (range(1, n-1) as i) { list(fibPrev, fib) = array(fib, fib + fibPrev); } return fib; }  ### Recursive function fibRec(n) { return n < 2 ? n : fibRec(n-1) + fibRec(n-2); }  ## PicoLisp ### Recursive (de fibo (N) (if (>= 2 N) 1 (+ (fibo (dec N)) (fibo (- N 2))) ) )  ### Recursive with Cache Using a recursive version doesn't need to be slow, as the following shows: (de fibo (N) (cache '(NIL) N # Use a cache to accelerate (if (>= 2 N) N (+ (fibo (dec N)) (fibo (- N 2))) ) ) ) (bench (fibo 1000))  Output: 0.012 sec -> 43466557686937456435688527675040625802564660517371780402481729089536555417949 05189040387984007925516929592259308032263477520968962323987332247116164299644090 6533187938298969649928516003704476137795166849228875  ### Iterative (de fib (N) (let (A 0 B 1) (do N (prog1 B (setq B (+ A B) A @)) ) ) )  ### Coroutines (co 'fibo (let (A 0 B 1) (yield 'ready) (while (yield (swap 'B (+ (swap 'A B) B)) ) ) ) ) (do 15 (printsp (yield 'next 'fibo)) ) (prinl) (yield NIL 'fibo)  {{out}} 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610  ## PIR Recursive: {{works with|Parrot|tested with 2.4.0}} .sub fib .param int n .local int nt .local int ft if n < 2 goto RETURNN nt = n - 1 ft = fib( nt ) dec nt nt = fib(nt) ft = ft + nt .return( ft ) RETURNN: .return( n ) end .end .sub main :main .local int counter .local int f counter=0 LOOP: if counter > 20 goto DONE f = fib(counter) print f print "\n" inc counter goto LOOP DONE: end .end  Iterative (stack-based): {{works with|Parrot|tested with 2.4.0}} .sub fib .param int n .local int counter .local int f .local pmc fibs .local int nmo .local int nmt fibs = new 'ResizableIntegerArray' if n == 0 goto RETURN0 if n == 1 goto RETURN1 push fibs, 0 push fibs, 1 counter = 2 FIBLOOP: if counter > n goto DONE nmo = pop fibs nmt = pop fibs f = nmo + nmt push fibs, nmt push fibs, nmo push fibs, f inc counter goto FIBLOOP RETURN0: .return( 0 ) end RETURN1: .return( 1 ) end DONE: f = pop fibs .return( f ) end .end .sub main :main .local int counter .local int f counter=0 LOOP: if counter > 20 goto DONE f = fib(counter) print f print "\n" inc counter goto LOOP DONE: end .end  ## Pike ### Iterative int fibIter(int n) { int fibPrev, fib, i; if (n < 2) { return 1; } fibPrev = 0; fib = 1; for (i = 1; i < n; i++) { int oldFib = fib; fib += fibPrev; fibPrev = oldFib; } return fib; }  ### Recursive int fibRec(int n) { if (n < 2) { return(1); } return( fib(n-2) + fib(n-1) ); }  ## PL/I /* Form the n-th Fibonacci number, n > 1. */ get list(n); f1 = 0; f2 = 1; do i = 2 to n; f3 = f1 + f2; put skip edit('fibo(',i,')=',f3)(a,f(5),a,f(5)); f1 = f2; f2 = f3; end;  ## PL/pgSQL ### Recursive CREATE OR REPLACE FUNCTION fib(n INTEGER) RETURNS INTEGER AS$$ BEGIN IF (n < 2) THEN RETURN n; END IF; RETURN fib(n - 1) + fib(n - 2); END; $$LANGUAGE plpgsql;  ### Calculated CREATE OR REPLACE FUNCTION fibFormula(n INTEGER) RETURNS INTEGER AS$$ BEGIN RETURN round(pow((pow(5, .5) + 1) / 2, n) / pow(5, .5)); END; $$LANGUAGE plpgsql;  ### Linear CREATE OR REPLACE FUNCTION fibLinear(n INTEGER) RETURNS INTEGER AS$$ DECLARE prevFib INTEGER := 0; fib INTEGER := 1; BEGIN IF (n < 2) THEN RETURN n; END IF; WHILE n > 1 LOOP SELECT fib, prevFib + fib INTO prevFib, fib; n := n - 1; END LOOP; RETURN fib; END; $$LANGUAGE plpgsql;  ### Tail recursive CREATE OR REPLACE FUNCTION fibTailRecursive(n INTEGER, prevFib INTEGER DEFAULT 0, fib INTEGER DEFAULT 1) RETURNS INTEGER AS$$ BEGIN IF (n = 0) THEN RETURN prevFib; END IF; RETURN fibTailRecursive(n - 1, fib, prevFib + fib); END; $$LANGUAGE plpgsql;  ## PL/SQL Create or replace Function fnu_fibonnaci(p_iNumber integer) return integer is nuFib integer; nuP integer; nuQ integer; Begin if p_iNumber is not null then if p_iNumber=0 then nuFib:=0; Elsif p_iNumber=1 then nuFib:=1; Else nuP:=0; nuQ:=1; For nuI in 2..p_iNumber loop nuFib:=nuP+nuQ; nuP:=nuQ; nuQ:=nuFib; End loop; End if; End if; return(nuFib); End fnu_fibonnaci;  ## Pop11 define fib(x); lvars a , b; 1 -> a; 1 -> b; repeat x - 1 times (a + b, b) -> (b, a); endrepeat; a; enddefine;  ## PostScript Enter the desired number for "n" and run through your favorite postscript previewer or send to your postscript printer: %!PS % We want the 'n'th fibonacci number /n 13 def % Prepare output canvas: /Helvetica findfont 20 scalefont setfont 100 100 moveto %define the function recursively: /fib { dup 3 lt { pop 1 } { dup 1 sub fib exch 2 sub fib add } ifelse } def (Fib$$) show n (....) cvs show ($$=) show n fib (.....) cvs show showpage  ## Potion ### Recursive Starts with int and upgrades on-the-fly to doubles. recursive = (n): if (n <= 1): 1. else: recursive (n - 1) + recursive (n - 2).. n = 40 ("fib(", n, ")= ", recursive (n), "\n") join print   recursive(40)= 165580141 real 0m2.851s  ### Iterative iterative = (n) : curr = 0 prev = 1 tmp = 0 n times: tmp = curr curr = curr + prev prev = tmp . curr .  ### Matrix based sqr = (x): x * x. # Based on the fact that # F2n = Fn(2Fn+1 - Fn) # F2n+1 = Fn ^2 + Fn+1 ^2 matrix = (n) : algorithm = (n) : "computes (Fn, Fn+1)" if (n < 2): return ((0, 1), (1, 1)) at(n). # n = e + {0, 1} q = algorithm(n / 2) # q = (Fe/2, Fe/2+1) q = (q(0) * (2 * q(1) - q(0)), sqr(q(0)) + sqr(q(1))) # q => (Fe, Fe+1) if (n % 2 == 1) : # q => (Fe+{0, 1}, Fe+1+{0,1}) = (Fn, Fn+1) q = (q(1), q(1) + q(0)) . q . algorithm(n)(0) .  ### Handling negative values fibonacci = (n) : myFavorite = matrix if (n >= 0) : myFavorite(n) . else : n = n * -1 if (n % 2 == 1) : myFavorite(n) . else : myFavorite(n) * -1 . . .  ## PowerBASIC {{trans|BASIC}} There seems to be a limitation (dare I say, bug?) in PowerBASIC regarding how large numbers are stored. 10E17 and larger get rounded to the nearest 10. For F(n), where ABS(n) > 87, is affected like this: actual: displayed: F(88) 1100087778366101931 1100087778366101930 F(89) 1779979416004714189 1779979416004714190 F(90) 2880067194370816120 2880067194370816120 F(91) 4660046610375530309 4660046610375530310 F(92) 7540113804746346429 7540113804746346430 powerbasic FUNCTION fibonacci (n AS LONG) AS QUAD DIM u AS LONG, a AS LONG, L0 AS LONG, outP AS QUAD STATIC fibNum() AS QUAD u = UBOUND(fibNum) a = ABS(n) IF u < 1 THEN REDIM fibNum(1) fibNum(1) = 1 u = 1 END IF SELECT CASE a CASE 0 TO 92 IF a > u THEN REDIM PRESERVE fibNum(a) FOR L0 = u + 1 TO a fibNum(L0) = fibNum(L0 - 1) + fibNum(L0 - 2) IF 88 = L0 THEN fibNum(88) = fibNum(88) + 1 NEXT END IF IF n < 0 THEN fibonacci = fibNum(a) * ((-1)^(a+1)) ELSE fibonacci = fibNum(a) END IF CASE ELSE 'Even without the above-mentioned bug, we're still limited to 'F(+/-92), due to data type limits. (F(93) = &hA94F AD42 221F 2702) ERROR 6 END SELECT END FUNCTION FUNCTION PBMAIN () AS LONG DIM n AS LONG #IF NOT %DEF(%PB_CC32) OPEN "out.txt" FOR OUTPUT AS 1 #ENDIF FOR n = -92 TO 92 #IF %DEF(%PB_CC32) PRINT STR(n); ": "; FORMAT(fibonacci(n), "#") #ELSE PRINT #1, STR(n) & ": " & FORMAT(fibonacci(n), "#") #ENDIF NEXT CLOSE END FUNCTION  ## PowerShell ### Iterative  function FibonacciNumber ( count ) { answer = @(0,1) while (answer.Length -le count) { answer += answer[-1] + answer[-2] } return answer }  An even shorter version that eschews function calls altogether:  count = 8 answer = @(0,1) 0..(count - answer.Length) | Foreach { answer += answer[-1] + answer[-2] } answer  ### Recursive function fib(n) { switch (n) { 0 { return 0 } 1 { return 1 } { _ -lt 0 } { return [Math]::Pow(-1, -n + 1) * (fib (-n)) } default { return (fib (n - 1)) + (fib (n - 2)) } } }  ## Processing {{trans|Java}} void setup() { size(400, 400); fill(255, 64); frameRate(2); } void draw() { int num = fibonacciNum(frameCount); println(frameCount, num); rect(0,0,num, num); if(frameCount==14) frameCount = -1; // restart } int fibonacciNum(int n) { return (n < 2) ? n : fibonacciNum(n - 1) + fibonacciNum(n - 2); }  On the nth frame, the nth Fibonacci number is printed to the console and a square of that size is drawn on the sketch surface. The sketch restarts to keep drawing within the window size. {{out}} 1 1 2 3 5 8 13 21 34 55 89 144 233 377  ## Prolog {{works with|SWI Prolog}} {{works with|GNU Prolog}} {{works with|YAP}}  fib(1, 1) :- !. fib(0, 0) :- !. fib(N, Value) :- A is N - 1, fib(A, A1), B is N - 2, fib(B, B1), Value is A1 + B1.  This naive implementation works, but is very slow for larger values of N. Here are some simple measurements (in SWI-Prolog): ?- time(fib(0,F)). % 2 inferences, 0.000 CPU in 0.000 seconds (88% CPU, 161943 Lips) F = 0. ?- time(fib(10,F)). % 265 inferences, 0.000 CPU in 0.000 seconds (98% CPU, 1458135 Lips) F = 55. ?- time(fib(20,F)). % 32,836 inferences, 0.016 CPU in 0.016 seconds (99% CPU, 2086352 Lips) F = 6765. ?- time(fib(30,F)). % 4,038,805 inferences, 1.122 CPU in 1.139 seconds (98% CPU, 3599899 Lips) F = 832040. ?- time(fib(40,F)). % 496,740,421 inferences, 138.705 CPU in 140.206 seconds (99% CPU, 3581264 Lips) F = 102334155.  As you can see, the calculation time goes up exponentially as N goes higher. ===Poor man's memoization=== {{works with|SWI Prolog}} {{works with|YAP}} {{works with|GNU Prolog}} The performance problem can be readily fixed by the addition of two lines of code (the first and last in this version): %:- dynamic fib/2. % This is ISO, but GNU doesn't like it. :- dynamic(fib/2). % Not ISO, but works in SWI, YAP and GNU unlike the ISO declaration. fib(1, 1) :- !. fib(0, 0) :- !. fib(N, Value) :- A is N - 1, fib(A, A1), B is N - 2, fib(B, B1), Value is A1 + B1, asserta((fib(N, Value) :- !)).  Let's take a look at the execution costs now: ?- time(fib(0,F)). % 2 inferences, 0.000 CPU in 0.000 seconds (90% CPU, 160591 Lips) F = 0. ?- time(fib(10,F)). % 37 inferences, 0.000 CPU in 0.000 seconds (96% CPU, 552610 Lips) F = 55. ?- time(fib(20,F)). % 41 inferences, 0.000 CPU in 0.000 seconds (96% CPU, 541233 Lips) F = 6765. ?- time(fib(30,F)). % 41 inferences, 0.000 CPU in 0.000 seconds (95% CPU, 722722 Lips) F = 832040. ?- time(fib(40,F)). % 41 inferences, 0.000 CPU in 0.000 seconds (96% CPU, 543572 Lips) F = 102334155.  In this case by asserting the new N,Value pairing as a rule in the database we're making the classic time/space tradeoff. Since the space costs are (roughly) linear by N and the time costs are exponential by N, the trade-off is desirable. You can see the poor man's memoizing easily: ?- listing(fib). :- dynamic fib/2. fib(40, 102334155) :- !. fib(39, 63245986) :- !. fib(38, 39088169) :- !. fib(37, 24157817) :- !. fib(36, 14930352) :- !. fib(35, 9227465) :- !. fib(34, 5702887) :- !. fib(33, 3524578) :- !. fib(32, 2178309) :- !. fib(31, 1346269) :- !. fib(30, 832040) :- !. fib(29, 514229) :- !. fib(28, 317811) :- !. fib(27, 196418) :- !. fib(26, 121393) :- !. fib(25, 75025) :- !. fib(24, 46368) :- !. fib(23, 28657) :- !. fib(22, 17711) :- !. fib(21, 10946) :- !. fib(20, 6765) :- !. fib(19, 4181) :- !. fib(18, 2584) :- !. fib(17, 1597) :- !. fib(16, 987) :- !. fib(15, 610) :- !. fib(14, 377) :- !. fib(13, 233) :- !. fib(12, 144) :- !. fib(11, 89) :- !. fib(10, 55) :- !. fib(9, 34) :- !. fib(8, 21) :- !. fib(7, 13) :- !. fib(6, 8) :- !. fib(5, 5) :- !. fib(4, 3) :- !. fib(3, 2) :- !. fib(2, 1) :- !. fib(1, 1) :- !. fib(0, 0) :- !. fib(A, D) :- B is A+ -1, fib(B, E), C is A+ -2, fib(C, F), D is E+F, asserta((fib(A, D):-!)).  All of the interim N/Value pairs have been asserted as facts for quicker future use, speeding up the generation of the higher Fibonacci numbers. ### Continuation passing style Works with SWI-Prolog and module lambda, written by Ulrich Neumerkel found there http://www.complang.tuwien.ac.at/ulrich/Prolog-inedit/lambda.pl :- use_module(lambda). fib(N, FN) :- cont_fib(N, _, FN, \_^Y^_^U^(U = Y)). cont_fib(N, FN1, FN, Pred) :- ( N < 2 -> call(Pred, 0, 1, FN1, FN) ; N1 is N - 1, P = \X^Y^Y^U^(U is X + Y), cont_fib(N1, FNA, FNB, P), call(Pred, FNA, FNB, FN1, FN) ).  ### With lazy lists Works with SWI-Prolog and others that support freeze/2. fib([0,1|X]) :- ffib(0,1,X). ffib(A,B,X) :- freeze(X, (C is A+B, X=[C|Y], ffib(B,C,Y)) ).  The predicate fib(Xs) unifies Xs with an infinite list whose values are the Fibonacci sequence. The list can be used like this: ?- fib(X), length(A,15), append(A,_,X), writeln(A). [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377]  ### Generators idiom take( 0, Next, Z-Z, Next). take( N, Next, [A|B]-Z, NZ):- N>0, !, next( Next, A, Next1), N1 is N-1, take( N1, Next1, B-Z, NZ). next( fib(A,B), A, fib(B,C)):- C is A+B. %% usage: ?- take(15, fib(0,1), _X-[], G), writeln(_X). %% [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377] %% G = fib(610, 987)  ### Yet another implementation One of my favorites; loosely similar to the first example, but without the performance penalty, and needs nothing special to implement. Not even a dynamic database predicate. Attributed to M.E. for the moment, but simply because I didn't bother to search for the many people who probably did it like this long before I did. If someone knows who came up with it first, please let us know. % Fibonacci sequence generator fib(C, [P,S], C, N) :- N is P + S. fib(C, [P,S], Cv, V) :- succ(C, Cn), N is P + S, !, fib(Cn, [S,N], Cv, V). fib(0, 0). fib(1, 1). fib(C, N) :- fib(2, [0,1], C, N). % Generate from 3rd sequence on  Looking at performance:  ?- time(fib(30,X)). % 86 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips) X = 832040 ?- time(fib(40,X)). % 116 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips) X = 102334155 ?- time(fib(100,X)). % 296 inferences, 0.000 CPU in 0.001 seconds (0% CPU, Infinite Lips) X = 354224848179261915075  What I really like about this one, is it is also a generator- i.e. capable of generating all the numbers in sequence needing no bound input variables or special Prolog predicate support (such as freeze/3 in the previous example): ?- time(fib(X,Fib)). % 0 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips) X = Fib, Fib = 0 ; % 1 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips) X = Fib, Fib = 1 ; % 3 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips) X = 2, Fib = 1 ; % 5 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips) X = 3, Fib = 2 ; % 5 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips) X = 4, Fib = 3 ; % 5 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips) X = Fib, Fib = 5 ; % 5 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips) X = 6, Fib = 8 ...etc.  It stays at 5 inferences per iteration after X=3. Also, quite useful:  ?- time(fib(100,354224848179261915075)). % 296 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips) true . ?- time(fib(X,354224848179261915075)). % 394 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips) X = 100 .  ## Pure ### Tail Recursive fib n = loop 0 1 n with loop a b n = if n==0 then a else loop b (a+b) (n-1); end;  ## PureBasic ### Macro based calculation Macro Fibonacci (n) Int((Pow(((1+Sqr(5))/2),n)-Pow(((1-Sqr(5))/2),n))/Sqr(5)) EndMacro  ### Recursive Procedure FibonacciReq(n) If n<2 ProcedureReturn n Else ProcedureReturn FibonacciReq(n-1)+FibonacciReq(n-2) EndIf EndProcedure  ===Recursive & optimized with a static hash table=== This will be much faster on larger n's, this as it uses a table to store known parts instead of recalculating them. On my machine the speedup compares to above code is Fib(n) Speedup 20 2 25 23 30 217 40 25847 46 1156741 Procedure Fibonacci(n) Static NewMap Fib.i() Protected FirstRecursion If MapSize(Fib())= 0 ; Init the hash table the first run Fib("0")=0: Fib("1")=1 FirstRecursion = #True EndIf If n >= 2 Protected.s s=Str(n) If Not FindMapElement(Fib(),s) ; Calculate only needed parts Fib(s)= Fibonacci(n-1)+Fibonacci(n-2) EndIf n = Fib(s) EndIf If FirstRecursion ; Free the memory when finalizing the first call ClearMap(Fib()) EndIf ProcedureReturn n EndProcedure  '''Example''' Fibonacci(0)= 0 Fibonacci(1)= 1 Fibonacci(2)= 1 Fibonacci(3)= 2 Fibonacci(4)= 3 Fibonacci(5)= 5 FibonacciReq(0)= 0 FibonacciReq(1)= 1 FibonacciReq(2)= 1 FibonacciReq(3)= 2 FibonacciReq(4)= 3 FibonacciReq(5)= 5 ## Purity The following takes a natural number and generates an initial segment of the Fibonacci sequence of that length:  data Fib1 = FoldNat < const (Cons One (Cons One Empty)), (uncurry Cons) . ((uncurry Add) . (Head, Head . Tail), id) >  This following calculates the Fibonacci sequence as an infinite stream of natural numbers:  type (Stream A?,,,Unfold) = gfix X. A? . X? data Fib2 = Unfold ((outl, (uncurry Add, outl))) ((curry id) One One)  As a histomorphism:  import Histo data Fib3 = Histo . Memoize < const One, (p1 => < const One, (p2 => Add (outl p1) (outl p2)). UnmakeCofree > (outr p1)) . UnmakeCofree >  ## Python ### Iterative positive and negative def fib(n,x=[0,1]): for i in range(abs(n)-1): x=[x[1],sum(x)] return x[1]*pow(-1,abs(n)-1) if n<0 else x[1] if n else 0 for i in range(-30,31): print fib(i),  Output:  -832040 514229 -317811 196418 -121393 75025 -46368 28657 -17711 10946 -6765 4181 -2584 1597 -987 610 -377 233 -144 89 -55 34 -21 13 -8 5 -3 2 -1 1 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040  ### Analytic Binet's formula: from math import * def analytic_fibonacci(n): sqrt_5 = sqrt(5); p = (1 + sqrt_5) / 2; q = 1/p; return int( (p**n + q**n) / sqrt_5 + 0.5 ) for i in range(1,31): print analytic_fibonacci(i),  Output:  1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040  ### Iterative def fibIter(n): if n < 2: return n fibPrev = 1 fib = 1 for num in xrange(2, n): fibPrev, fib = fib, fib + fibPrev return fib  ### Recursive def fibRec(n): if n < 2: return n else: return fibRec(n-1) + fibRec(n-2)  ### Recursive with Memoization def fibMemo(): pad = {0:0, 1:1} def func(n): if n not in pad: pad[n] = func(n-1) + func(n-2) return pad[n] return func fm = fibMemo() for i in range(1,31): print fm(i),  Output:  1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040  ===Better Recursive doesn't need Memoization=== The recursive code as written two sections above is incredibly slow and inefficient due to the nested recursion calls. Although the memoization above makes the code run faster, it is at the cost of extra memory use. The below code is syntactically recursive but actually encodes the efficient iterative process, and thus doesn't require memoization: def fibFastRec(n): def fib(prvprv, prv, c): if c < 1: return prvprv else: return fib(prv, prvprv + prv, c - 1) return fib(0, 1, n)  However, although much faster and not requiring memory, the above code can only work to a limited 'n' due to the limit on stack recursion depth by Python; it is better to use the iterative code above or the generative one below. ### Generative def fibGen(n): a, b = 0, 1 while n>0: yield a a, b, n = b, a+b, n-1  ### =Example use=  >>> [i for i in fibGen(11)] [0,1,1,2,3,5,8,13,21,34,55]  ===Matrix-Based=== Translation of the matrix-based approach used in F#.  def prevPowTwo(n): 'Gets the power of two that is less than or equal to the given input' if ((n & -n) == n): return n else: n -= 1 n |= n >> 1 n |= n >> 2 n |= n >> 4 n |= n >> 8 n |= n >> 16 n += 1 return (n/2) def crazyFib(n): 'Crazy fast fibonacci number calculation' powTwo = prevPowTwo(n) q = r = i = 1 s = 0 while(i < powTwo): i *= 2 q, r, s = q*q + r*r, r * (q + s), (r*r + s*s) while(i < n): i += 1 q, r, s = q+r, q, r return q  ### Large step recurrence This is much faster for a single, large value of n: def fib(n, c={0:1, 1:1}): if n not in c: x = n // 2 c[n] = fib(x-1) * fib(n-x-1) + fib(x) * fib(n - x) return c[n] fib(10000000) # calculating it takes a few seconds, printing it takes eons  ### Same as above but slightly faster Putting the dictionary outside the function makes this about 2 seconds faster, could just make a wrapper: F = {0: 0, 1: 1, 2: 1} def fib(n): if n in F: return F[n] f1 = fib(n // 2 + 1) f2 = fib((n - 1) // 2) F[n] = (f1 * f1 + f2 * f2 if n & 1 else f1 * f1 - f2 * f2) return F[n]  ### Generative with Recursion This can get very slow and uses a lot of memory. Can be sped up by caching the generator results. def fib(): """Yield fib[n+1] + fib[n]""" yield 1 # have to start somewhere lhs, rhs = fib(), fib() yield next(lhs) # move lhs one iteration ahead while True: yield next(lhs)+next(rhs) f=fib() print [next(f) for _ in range(9)]  Output: [1, 1, 2, 3, 5, 8, 13, 21, 34]  '''Another version of recursive generators solution, starting from 0''' from itertools import islice def fib(): yield 0 yield 1 a, b = fib(), fib() next(b) while True: yield next(a)+next(b) print(tuple(islice(fib(), 10)))  ### As a scan or a fold ### =itertools.accumulate= The Fibonacci series can be defined quite simply and efficiently as a scan or accumulation, in which the accumulator is a pair of the two last numbers. {{Works with|Python|3.7}} '''Fibonacci accumulation''' from itertools import (accumulate, chain) # fibs :: Integer :: [Integer] def fibs(n): '''An accumulation of the first n integers in the Fibonacci series. The accumulator is a pair of the two preceding numbers. ''' def go(ab, _): a, b = ab return (b, a + b) return [xy[1] for xy in accumulate( chain( [(0, 1)], range(1, n) ), go )] # MAIN --- if __name__ == '__main__': print( 'First twenty: ' + repr( fibs(20) ) )  {{Out}} First twenty: [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765]  ### =functools.reduce= A fold can be understood as an amnesic scan, and functools.reduce can provide a useful and efficient re-write of the scanning version above, if we only need the Nth term in the series: {{Works with|Python|3.7}} '''Nth Fibonacci term (by folding)''' from functools import (reduce) # nthFib :: Integer -> Integer def nthFib(n): '''Nth integer in the Fibonacci series.''' def go(ab, _): a, b = ab return (b, a + b) return reduce(go, range(1, n), (0, 1))[1] # MAIN --- if __name__ == '__main__': print( '1000th term: ' + repr( nthFib(1000) ) )  {{Out}} 1000th term: 43466557686937456435688527675040625802564660517371780402481729089536555417949051890403879840079255169295922593080322634775209689623239873322471161642996440906533187938298969649928516003704476137795166849228875  ## Qi ### Recursive  (define fib 0 -> 0 1 -> 1 N -> (+ (fib-r (- N 1)) (fib-r (- N 2))))  ### Iterative  (define fib-0 V2 V1 0 -> V2 V2 V1 N -> (fib-0 V1 (+ V2 V1) (1- N))) (define fib N -> (fib-0 0 1 N))  ## R ### Iterative positive and negative fib=function(n,x=c(0,1)) { if (abs(n)>1) for (i in seq(abs(n)-1)) x=c(x[2],sum(x)) if (n<0) return(x[2]*(-1)^(abs(n)-1)) else if (n) return(x[2]) else return(0) } sapply(seq(-31,31),fib)  Output:  [1] 1346269 -832040 514229 -317811 196418 -121393 75025 -46368 28657 [10] -17711 10946 -6765 4181 -2584 1597 -987 610 -377 [19] 233 -144 89 -55 34 -21 13 -8 5 [28] -3 2 -1 1 0 1 1 2 3 [37] 5 8 13 21 34 55 89 144 233 [46] 377 610 987 1597 2584 4181 6765 10946 17711 [55] 28657 46368 75025 121393 196418 317811 514229 832040 1346269  ### Other methods # recursive recfibo <- function(n) { if ( n < 2 ) n else Recall(n-1) + Recall(n-2) } # print the first 21 elements print.table(lapply(0:20, recfibo)) # iterative iterfibo <- function(n) { if ( n < 2 ) n else { f <- c(0, 1) for (i in 2:n) { t <- f[2] f[2] <- sum(f) f[1] <- t } f[2] } } print.table(lapply(0:20, iterfibo)) # iterative but looping replaced by map-reduce'ing funcfibo <- function(n) { if (n < 2) n else { generator <- function(f, ...) { c(f[2], sum(f)) } Reduce(generator, 2:n, c(0,1))[2] } } print.table(lapply(0:20, funcfibo))  Note that an idiomatic way to implement such low level, basic arithmethic operations in R is to implement them C and then call the compiled code. {{out}} All three solutions print  [1] 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 [16] 610 987 1597 2584 4181 6765  ## Ra  class FibonacciSequence **Prints the nth fibonacci number** on start args := program arguments if args empty print .fibonacci(8) else try print .fibonacci(integer.parse(args[0])) catch FormatException print to Console.error made !, "Input must be an integer" exit program with error code catch OverflowException print to Console.error made !, "Number too large" exit program with error code define fibonacci(n as integer) as integer is shared **Returns the nth fibonacci number** test assert fibonacci(0) = 0 assert fibonacci(1) = 1 assert fibonacci(2) = 1 assert fibonacci(3) = 2 assert fibonacci(4) = 3 assert fibonacci(5) = 5 assert fibonacci(6) = 8 assert fibonacci(7) = 13 assert fibonacci(8) = 21 body a, b := 0, 1 for n a, b := b, a + b return a  ## Racket ### Tail Recursive  (define (fib n) (let loop ((cnt 0) (a 0) (b 1)) (if (= n cnt) a (loop (+ cnt 1) b (+ a b)))))   (define (fib n (a 0) (b 1)) (if (< n 2) 1 (+ a (fib (- n 1) b (+ a b)))))  ### Matrix Form  #lang racket (require math/matrix) (define (fibmat n) (matrix-ref (matrix-expt (matrix ([1 1] [1 0])) n) 1 0)) (fibmat 1000)  ## Retro ### Recursive : fib ( n-m ) dup [ 0 = ] [ 1 = ] bi or if; [ 1- fib ] sip [ 2 - fib ] do + ;  ### Iterative : fib ( n-N ) [ 0 1 ] dip [ over + swap ] times drop ;  ## REXX With 210,000 numeric decimal digits, this REXX program can handle Fibonacci numbers past one million. [Generally speaking, some REXX interpreters can handle up to around eight million decimal digits.] This version of the REXX program can also handle ''negative'' Fibonacci numbers. /*REXX program calculates the Nth Fibonacci number, N can be zero or negative. */ numeric digits 210000 /*be able to handle ginormous numbers. */ parse arg x y . /*allow a single number or a range. */ if x=='' | x=="," then do; x=-40; y=+40; end /*No input? Then use range -40 ──► +40*/ if y=='' | y=="," then y=x /*if only one number, display fib(X).*/ w= max(length(x), length(y) ) /*W: used for making formatted output.*/ fw= 10 /*Minimum maximum width. Sounds ka─razy*/ do j=x to y; q= fib(j) /*process all of the Fibonacci requests*/ L= length(q) /*obtain the length (decimal digs) of Q*/ fw= max(fw, L) /*fib number length, or the max so far.*/ say 'Fibonacci('right(j,w)") = " right(q,fw) /*right justify Q.*/ if L>10 then say 'Fibonacci('right(j, w)") has a length of" L end /*j*/ /* [↑] list a Fib. sequence of x──►y */ exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ fib: procedure; parse arg n; an= abs(n) /*use │n│ (the absolute value of N).*/ a= 0; b= 1; if an<2 then return an /*handle two special cases: zero & one.*/ /* [↓] this method is non─recursive. */ do k=2 to an; = a+b; a= b; b= /*sum the numbers up to │n│ */ end /*k*/ /* [↑] (only positive Fibs nums used).*/ /* [↓] an//2 [same as] (an//2==1).*/ if n>0 | an//2 then return /*Positive or even? Then return sum. */ return - /*Negative and odd? Return negative sum*/  {{out|output|text= when using the default inputs:}} Fibonacci(-40) = -102334155 Fibonacci(-39) = 63245986 Fibonacci(-38) = -39088169 Fibonacci(-37) = 24157817 Fibonacci(-36) = -14930352 Fibonacci(-35) = 9227465 Fibonacci(-34) = -5702887 Fibonacci(-33) = 3524578 Fibonacci(-32) = -2178309 Fibonacci(-31) = 1346269 Fibonacci(-30) = -832040 Fibonacci(-29) = 514229 Fibonacci(-28) = -317811 Fibonacci(-27) = 196418 Fibonacci(-26) = -121393 Fibonacci(-25) = 75025 Fibonacci(-24) = -46368 Fibonacci(-23) = 28657 Fibonacci(-22) = -17711 Fibonacci(-21) = 10946 Fibonacci(-20) = -6765 Fibonacci(-19) = 4181 Fibonacci(-18) = -2584 Fibonacci(-17) = 1597 Fibonacci(-16) = -987 Fibonacci(-15) = 610 Fibonacci(-14) = -377 Fibonacci(-13) = 233 Fibonacci(-12) = -144 Fibonacci(-11) = 89 Fibonacci(-10) = -55 Fibonacci( -9) = 34 Fibonacci( -8) = -21 Fibonacci( -7) = 13 Fibonacci( -6) = -8 Fibonacci( -5) = 5 Fibonacci( -4) = -3 Fibonacci( -3) = 2 Fibonacci( -2) = -1 Fibonacci( -1) = 1 Fibonacci( 0) = 0 Fibonacci( 1) = 1 Fibonacci( 2) = 1 Fibonacci( 3) = 2 Fibonacci( 4) = 3 Fibonacci( 5) = 5 Fibonacci( 6) = 8 Fibonacci( 7) = 13 Fibonacci( 8) = 21 Fibonacci( 9) = 34 Fibonacci( 10) = 55 Fibonacci( 11) = 89 Fibonacci( 12) = 144 Fibonacci( 13) = 233 Fibonacci( 14) = 377 Fibonacci( 15) = 610 Fibonacci( 16) = 987 Fibonacci( 17) = 1597 Fibonacci( 18) = 2584 Fibonacci( 19) = 4181 Fibonacci( 20) = 6765 Fibonacci( 21) = 10946 Fibonacci( 22) = 17711 Fibonacci( 23) = 28657 Fibonacci( 24) = 46368 Fibonacci( 25) = 75025 Fibonacci( 26) = 121393 Fibonacci( 27) = 196418 Fibonacci( 28) = 317811 Fibonacci( 29) = 514229 Fibonacci( 30) = 832040 Fibonacci( 31) = 1346269 Fibonacci( 32) = 2178309 Fibonacci( 33) = 3524578 Fibonacci( 34) = 5702887 Fibonacci( 35) = 9227465 Fibonacci( 36) = 14930352 Fibonacci( 37) = 24157817 Fibonacci( 38) = 39088169 Fibonacci( 39) = 63245986 Fibonacci( 40) = 102334155  '''output''' when the following was used as input: 10000 txt Fibonacci(10000) = 3364476487643178326662161200510754331030214846068006390656476997468008144216666236815559551363373402558206533268083615937373479048386526826304089246305643188735454436955982749160660209988418393386465273130008883026923567361313511757929743785441375213052050434770160226475831890652789085515436615958298727968298751063120057542878345321551510387081829896979161312785626503319548714021428753269818796204693609787990035096230229102636813149319527563022783762844154036058440257211433496118002309120828704608892396232883546150577658327125254609359112820392528539343462090424524892940390 170623388899108584106518317336043747073790855263176432573399371287193758774689747992630583706574283016163740896917842637862421283525811282051637029808933209990570792006436742620238978311147005407499845925036063356093388383192338678305613643535189213327973290813373264265263398976392272340788292817795358057099369104917547080893184105614632233821746563732124822638309210329770164805472624384237486241145309381220656491403275108664339451751216152654536133311131404243685480510676584349352383695965342807176877532834823434555736671973139274627362910821067928078471803532913117677892465908993863545932789 452377767440619224033763867400402133034329749690202832814593341882681768389307200363479562311710310129195316979460763273758925353077255237594378843450406771555577905645044301664011946258097221672975861502696844314695203461493229110597067624326851599283470989128470674086200858713501626031207190317208609408129832158107728207635318662461127824553720853236530577595643007251774431505153960090516860322034916322264088524885243315805153484962243484829938090507048348244932745373262456775587908918719080366205800959474315005240253270974699531877072437682590741993963226598414749819360928522394503970716544 3156421328157688908058783183404917434556270520223564846495196112460268313970975069382648706613264507665074611512677522748621598642530711298441182622661057163515069260029861704945425047491378115154139941550671256271197133252763631939606902895650288268608362241082050562430701794976171121233066073310059947366875 Fibonacci(10000) has a length of 2090 decimal digits  ## Rockstar ===Iterative (minimized)=== Rockstar Fibonacci takes Number FNow is 0 FNext is 1 While FNow is less than Number Say FNow Put FNow into Temp Put FNow into FNext Put FNext plus Temp into FNext Say Fibonacci taking 1000 (prints out highest number in Fibonacci sequence less than 1000)  ===Iterative (idiomatic)=== Rockstar Love takes Time My love was addictions Put my love into your heart Build it up Until my love is as strong as Time Whisper my love Put my love into a river Put your heart into my love Put it with a river into your heart Shout; Love taking 1000 (years, years)  The semicolon and the comment (years, years) in this version are there only for poetic effect ## Ring ring give n x = fib(n) see n + " Fibonacci is : " + x func fib nr if nr = 0 return 0 ok if nr = 1 return 1 ok if nr > 1 return fib(nr-1) + fib(nr-2) ok  ## Ruby ### Iterative ruby def fib(n, sequence=[1]) n.times do current_number, last_number = sequence.last(2) sequence << current_number + (last_number or 0) end sequence.last end  ### Recursive ruby def fib(n, sequence=[1]) return sequence.last if n == 0 current_number, last_number = sequence.last(2) sequence << current_number + (last_number or 0) fib(n-1, sequence) end  ### Recursive with Memoization ruby # Use the Hash#default_proc feature to # lazily calculate the Fibonacci numbers. fib = Hash.new do |f, n| f[n] = if n <= -2 (-1)**(n + 1) * f[n.abs] elsif n <= 1 n.abs else f[n - 1] + f[n - 2] end end # examples: fib[10] => 55, fib[-10] => (-55/1)  ### Matrix ruby require 'matrix' # To understand why this matrix is useful for Fibonacci numbers, remember # that the definition of Matrix.**2 for any Matrix[[a, b], [c, d]] is # is [[a*a + b*c, a*b + b*d], [c*a + d*b, c*b + d*d]]. In other words, the # lower right element is computing F(k - 2) + F(k - 1) every time M is multiplied # by itself (it is perhaps easier to understand this by computing M**2, 3, etc, and # watching the result march up the sequence of Fibonacci numbers). M = Matrix[[0, 1], [1,1]] # Matrix exponentiation algorithm to compute Fibonacci numbers. # Let M be Matrix [[0, 1], [1, 1]]. Then, the lower right element of M**k is # F(k + 1). In other words, the lower right element of M is F(2) which is 1, and the # lower right element of M**2 is F(3) which is 2, and the lower right element # of M**3 is F(4) which is 3, etc. # # This is a good way to compute F(n) because the Ruby implementation of Matrix.**(n) # uses O(log n) rather than O(n) matrix multiplications. It works by squaring squares # ((m**2)**2)... as far as possible # and then multiplying that by by M**(the remaining number of times). E.g., to compute # M**19, compute partial = ((M**2)**2) = M**16, and then compute partial*(M**3) = M**19. # That's only 5 matrix multiplications of M to compute M*19. def self.fib_matrix(n) return 0 if n <= 0 # F(0) return 1 if n == 1 # F(1) # To get F(n >= 2), compute M**(n - 1) and extract the lower right element. return CS::lower_right(M**(n - 1)) end # Matrix utility to return # the lower, right-hand element of a given matrix. def self.lower_right matrix return nil if matrix.row_size == 0 return matrix[matrix.row_size - 1, matrix.column_size - 1] end  ### Generative ruby fib = Enumerator.new do |y| f0, f1 = 0, 1 loop do y << f0 f0, f1 = f1, f0 + f1 end end  Usage: txt p fib.lazy.drop(8).next # => 21  {{works with|Ruby|1.9}} "Fibers are primitives for implementing light weight cooperative concurrency in Ruby. Basically they are a means of creating code blocks that can be paused and resumed, much like threads. The main difference is that they are never preempted and that the scheduling must be done by the programmer and not the VM." [http://www.ruby-doc.org/ruby-1.9/classes/Fiber.html] ruby fib = Fiber.new do a,b = 0,1 loop do Fiber.yield a a,b = b,a+b end end 9.times {puts fib.resume}  using a lambda ruby def fib_gen a, b = 1, 1 lambda {ret, a, b = a, b, a+b; ret} end  txt irb(main):034:0> fg = fib_gen => # irb(main):035:0> 9.times { puts fg.call} 1 1 2 3 5 8 13 21 34 => 9  ===Binet's Formula=== ruby def fib phi = (1 + Math.sqrt(5)) / 2 ((phi**self - (-1 / phi)**self) / Math.sqrt(5)).to_i end  txt 1.9.3p125 :001 > def fib 1.9.3p125 :002?> phi = (1 + Math.sqrt(5)) / 2 1.9.3p125 :003?> ((phi**self - (-1 / phi)**self) / Math.sqrt(5)).to_i 1.9.3p125 :004?> end => nil 1.9.3p125 :005 > (0..10).map(&:fib) => [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]  ## Run BASIC runbasic for i = 0 to 10 print i;" ";fibR(i);" ";fibI(i) next i end function fibR(n) if n < 2 then fibR = n else fibR = fibR(n-1) + fibR(n-2) end function function fibI(n) b = 1 for i = 1 to n t = a + b a = b b = t next i fibI = a end function  ## Rust ### Iterative rust use std::mem; fn main() { let mut prev = 0; // Rust needs this type hint for the checked_add method let mut curr = 1usize; while let Some(n) = curr.checked_add(prev) { prev = curr; curr = n; println!("{}", n); } }  ### Recursive rust use std::mem; fn main() { fibonacci(0,1); } fn fibonacci(mut prev: usize, mut curr: usize) { mem::swap(&mut prev, &mut curr); if let Some(n) = curr.checked_add(prev) { println!("{}", n); fibonacci(prev, n); } }  ===Recursive (with pattern matching)=== rust fn fib(n: u32) -> u32 { match n { 0 => 0, 1 => 1, n => fib(n - 1) + fib(n - 2), } }  ===Tail recursive (with pattern matching)=== rust fn fib_tail_recursive(nth: usize) -> usize { fn fib_tail_iter(n: usize, prev_fib: usize, fib: usize) -> usize { match n { 0 => prev_fib, n => fib_tail_iter(n - 1, fib, prev_fib + fib), } } fib_tail_iter(nth, 0, 1) }  ### Analytic This uses a feature from nightly Rust which makes it possible to (cleanly) return an iterator without the additional overhead of putting it on the heap. In stable Rust, we'd need to return a Box> which has the cost of an additional allocation and the overhead of dynamic dispatch. The version below does not require the use of the heap and is done entirely through static dispatch. rust #![feature(conservative_impl_trait)] fn main() { for num in fibonacci_gen(10) { println!("{}", num); } } fn fibonacci_gen(terms: i32) -> impl Iterator { let sqrt_5 = 5.0f64.sqrt(); let p = (1.0 + sqrt_5) / 2.0; let q = 1.0/p; (1..terms).map(move |n| ((p.powi(n) + q.powi(n)) / sqrt_5 + 0.5) as u64) }  ### Using an Iterator Iterators are very idiomatic in rust, though they may be overkill for such a simple problem. rust use std::mem; struct Fib { prev: usize, curr: usize, } impl Fib { fn new() -> Self { Fib {prev: 0, curr: 1} } } impl Iterator for Fib { type Item = usize; fn next(&mut self) -> Option{ mem::swap(&mut self.curr, &mut self.prev); self.curr.checked_add(self.prev).map(|n| { self.curr = n; n }) } } fn main() { for num in Fib::new() { println!("{}", num); } }  ## SAS ### Iterative This code builds a table fib holding the first few values of the Fibonacci sequence. sas data fib; a=0; b=1; do n=0 to 20; f=a; output; a=b; b=f+a; end; keep n f; run;  ### Naive recursive This code provides a simple example of defining a function and using it recursively. One of the members of the sequence is written to the log. sas options cmplib=work.f; proc fcmp outlib=work.f.p; function fib(n); if n = 0 or n = 1 then return(1); else return(fib(n - 2) + fib(n - 1)); endsub; run; data _null_; x = fib(5); put 'fib(5) = ' x; run;  ## Sather The implementations use the arbitrary precision class INTI. sather class MAIN is -- RECURSIVE -- fibo(n :INTI):INTI pre n >= 0 is if n < 2.inti then return n; end; return fibo(n - 2.inti) + fibo(n - 1.inti); end; -- ITERATIVE -- fibo_iter(n :INTI):INTI pre n >= 0 is n3w :INTI; if n < 2.inti then return n; end; last ::= 0.inti; this ::= 1.inti; loop (n - 1.inti).times!; n3w := last + this; last := this; this := n3w; end; return this; end; main is loop i ::= 0.upto!(16); #OUT + fibo(i.inti) + " "; #OUT + fibo_iter(i.inti) + "\n"; end; end; end;  =={{header|S-BASIC}}== Note that the 23rd Fibonacci number (=28657) is the largest that can be generated without overflowing S-BASIC's integer data type. basic rem - iterative function to calculate nth fibonacci number function fibonacci(n = integer) = integer var f, i, p1, p2 = integer p1 = 0 p2 = 1 if n = 0 then f = 0 else for i = 1 to n f = p1 + p2 p2 = p1 p1 = f next i end = f rem - exercise the function var i = integer for i = 0 to 10 print fibonacci(i); next i end  {{out}} txt 0 1 1 2 3 5 8 13 21 34 55  ## Scala ### Recursive scala def fib(i:Int):Int = i match{ case 0 => 0 case 1 => 1 case _ => fib(i-1) + fib(i-2) }  ### Lazy sequence scala lazy val fib: Stream[Int] = 0 #:: 1 #:: fib.zip(fib.tail).map{case (a,b) => a + b}  ### Tail recursive scala def fib(x:Int, prev: BigInt = 0, next: BigInt = 1):BigInt = x match { case 0 => prev case _ => fib(x-1, next, next + prev) }  ### foldLeft scala // Fibonacci using BigInt with Stream.foldLeft optimized for GC (Scala v2.9 and above) // Does not run out of memory for very large Fibonacci numbers def fib(n:Int) = { def series(i:BigInt,j:BigInt):Stream[BigInt] = i #:: series(j, i+j) series(1,0).take(n).foldLeft(BigInt("0"))(_+_) } // Small test (0 to 13) foreach {n => print(fib(n).toString + " ")} // result: 0 1 1 2 3 5 8 13 21 34 55 89 144 233  ### Iterator scala val it = Iterator.iterate((0,1)){case (a,b) => (b,a+b)}.map(_._1) //example: println(it.take(13).mkString(",")) //prints: 0,1,1,2,3,5,8,13,21,34,55,89,144  ## Scheme ### Iterative scheme (define (fib-iter n) (do ((num 2 (+ num 1)) (fib-prev 1 fib) (fib 1 (+ fib fib-prev))) ((>= num n) fib)))  ### Recursive scheme (define (fib-rec n) (if (< n 2) n (+ (fib-rec (- n 1)) (fib-rec (- n 2)))))  This version is tail recursive: scheme (define (fib n) (let loop ((a 0) (b 1) (n n)) (if (= n 0) a (loop b (+ a b) (- n 1)))))  ### Recursive Sequence Generator Although the tail recursive version above is quite efficient, it only generates the final nth Fibonacci number and not the sequence up to that number without wasteful repeated calls to the procedure/function. The following procedure generates the sequence of Fibonacci numbers using a simplified version of a lazy list/stream - since no memoization is requried, it just implements future values by using a zero parameter lambda "thunk" with a closure containing the last and the pre-calculated next value of the sequence; in this way it uses almost no memory during the sequence generation other than as required for the last and the next values of the sequence (note that the test procedure does not generate a linked list to contain the elements of the sequence to show, but rather displays each one by one in sequence): scheme (define (fib) (define (nxt lv nv) (cons nv (lambda () (nxt nv (+ lv nv))))) (cons 0 (lambda () (nxt 0 1)))) ;;; test... (define (show-stream-take n strm) (define (shw-nxt n strm) (begin (display (car strm)) (if (> n 1) (begin (display " ") (shw-nxt (- n 1) ((cdr strm)))) (display ")")))) (begin (display "(") (shw-nxt n strm))) (show-stream-take 30 (fib))  {{output}} txt (0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229)  ### Dijkstra Algorithm scheme ;;; Fibonacci numbers using Edsger Dijkstra's algorithm ;;; http://www.cs.utexas.edu/users/EWD/ewd06xx/EWD654.PDF (define (fib n) (define (fib-aux a b p q count) (cond ((= count 0) b) ((even? count) (fib-aux a b (+ (* p p) (* q q)) (+ (* q q) (* 2 p q)) (/ count 2))) (else (fib-aux (+ (* b q) (* a q) (* a p)) (+ (* b p) (* a q)) p q (- count 1))))) (fib-aux 1 0 0 1 n))  ## Scilab clear n=46 f1=0; f2=1 printf("fibo(%d)=%d\n",0,f1) printf("fibo(%d)=%d\n",1,f2) for i=2:n f3=f1+f2 printf("fibo(%d)=%d\n",i,f3) f1=f2 f2=f3 end  {{out}} txt ... fibo(43)=433494437 fibo(44)=701408733 fibo(45)=1134903170 fibo(46)=1836311903  ## sed sed #!/bin/sed -f # First we need to convert each number into the right number of ticks # Start by marking digits s/[0-9]/<&/g # We have to do the digits manually. s/0//g; s/1/|/g; s/2/||/g; s/3/|||/g; s/4/||||/g; s/5/|||||/g s/6/||||||/g; s/7/|||||||/g; s/8/||||||||/g; s/9/|||||||||/g # Multiply by ten for each digit from the front. :tens s/|= 2 ticks into two of n-1, with a mark between s/|$$|\+$$/\1-\1/g # Convert the previous mark and the first tick after it to a different mark # giving us n-1+n-2 marks. s/-|/+/g # Jump back unless we're done. t split # Get rid of the pluses, we're done with them. s/+//g # Convert back to digits :back s/||||||||||/ 2 then result := fib(pred(number)) + fib(number - 2); elsif number = 0 then result := 0; end if; end func;  Original source: [http://seed7.sourceforge.net/algorith/math.htm#fib] ### Iterative This funtion uses a bigInteger result: seed7 const func bigInteger: fib (in integer: number) is func result var bigInteger: result is 1_; local var integer: i is 0; var bigInteger: a is 0_; var bigInteger: c is 0_; begin for i range 1 to pred(number) do c := a; a := result; result +:= c; end for; end func;  Original source: [http://seed7.sourceforge.net/algorith/math.htm#iterative_fib] ## SequenceL ### Recursive sequencel fibonacci(n) := n when n < 2 else fibonacci(n - 1) + fibonacci(n - 2);  Based on: [https://www.youtube.com/watch?v=5JVC5dDtnyg] ### Tail Recursive sequencel fibonacci(n) := fibonacciHelper(0, 1, n); fibonacciHelper(prev, next, n) := prev when n < 1 else next when n = 1 else fibonacciHelper(next, next + prev, n - 1);  ### Matrix sequencel fibonacci(n) := fibonacciHelper([[1,0],[0,1]], n); fibonacciHelper(M(2), n) := let N := [[1,1],[1,0]]; in M[1,1] when n <= 1 else fibonacciHelper(matmul(M, N), n - 1); matmul(A(2), B(2)) [i,j] := sum( A[i,all] * B[all,j] );  Based on the C# version: [http://rosettacode.org/wiki/Fibonacci_sequence#C.23] Using the SequenceL Matrix Multiply solution: [http://rosettacode.org/wiki/Matrix_multiplication#SequenceL] ## SETL setl Print out the first ten Fibonacci numbers This uses Set Builder Notation, it roughly means 'collect fib(n) forall n in {0,1,2,3,4,5,6,7,8,9,10}' print({fib(n) : n in {0..10}}); Iterative Fibonacci function proc fib(n); A := 0; B := 1; C := n; for i in {0..n} loop C := A + B; A := B; B := C; end loop; return C; end proc;  ## Shen Shen (define fib 0 -> 0 1 -> 1 N -> (+ (fib (+ N 1)) (fib (+ N 2))) where (< N 0) N -> (+ (fib (- N 1)) (fib (- N 2))))  ## Sidef ### Iterative ruby func fib_iter(n) { var (a, b) = (0, 1) { (a, b) = (b, a+b) } * n return a }  ### Recursive ruby func fib_rec(n) { n < 2 ? n : (__FUNC__(n-1) + __FUNC__(n-2)) }  ### Recursive with memoization ruby func fib_mem (n) is cached { n < 2 ? n : (__FUNC__(n-1) + __FUNC__(n-2)) }  ===Closed-form=== ruby func fib_closed(n) { define S = (1.25.sqrt + 0.5) define T = (-S + 1) (S**n - T**n) / (-T + S) -> round }  ===Built-in=== ruby say fib(12) #=> 144  ## Simula Straightforward iterative implementation. simula INTEGER PROCEDURE fibonacci(n); INTEGER n; BEGIN INTEGER lo, hi, temp, i; lo := 0; hi := 1; FOR i := 1 STEP 1 UNTIL n - 1 DO BEGIN temp := hi; hi := hi + lo; lo := temp END; fibonacci := hi END;  ## SkookumScript ===Built-in=== SkookumScript's Integer class has a fast built-in fibonnaci() method. javascript>42.fibonacci42.fibonacci form as built-in form above. javascript // Assuming code is in Integer.fibonacci() method () Integer [ if this < 2 [this] else [[this - 1].fibonacci + [this - 2].fibonacci] ]  Recursive procedure in fibonacci(42) form. javascript // Assuming in fibonacci(n) procedure (Integer n) Integer [ if n < 2 [n] else [fibonacci(n - 1) + fibonacci(n - 2)] ]  ### Iterative Iterative method in 42.fibonacci form. javascript // Assuming code is in Integer.fibonacci() method () Integer [ if this < 2 [this] else [ !prev: 1 !next: 1 2.to_pre this [ !sum : prev + next prev := next next := sum ] next ] ]  Optimized iterative method in 42.fibonacci form. Though the best optimiation is to write it in C++ as with the built-in form that comes with SkookumScript. javascript // Bind : is faster than assignment := // loop is faster than to_pre (which uses a closure) () Integer [ if this < 2 [this] else [ !prev: 1 !next: 1 !sum !count: this - 2 loop [ if count = 0 [exit] count-- sum : prev + next prev : next next : sum ] next ] ]  ## Slate slate n@(Integer traits) fib [ n <= 0 ifTrue: [^ 0]. n = 1 ifTrue: [^ 1]. (n - 1) fib + (n - 2) fib ]. slate[15]> 10 fib = 55. True  ## Smalltalk smalltalk |fibo| fibo := [ :i | |ac t| ac := Array new: 2. ac at: 1 put: 0 ; at: 2 put: 1. ( i < 2 ) ifTrue: [ ac at: (i+1) ] ifFalse: [ 2 to: i do: [ :l | t := (ac at: 2). ac at: 2 put: ( (ac at: 1) + (ac at: 2) ). ac at: 1 put: t ]. ac at: 2. ] ]. 0 to: 10 do: [ :i | (fibo value: i) displayNl ]  ## smart BASIC The Iterative method is slow (relatively) and the Recursive method doubly so since it references the Iterative function twice. The N-th Term (fibN) function is much faster as it utilizes Binet's Formula. fibR: Fibonacci Recursive fibI: Fibonacci Iterative fibN: Fibonacci N-th Term qbasic FOR i = 0 TO 15 PRINT fibR(i),fibI(i),fibN(i) NEXT i /* Recursive Method */ DEF fibR(n) IF n <= 1 THEN fibR = n ELSE fibR = fibR(n-1) + fibR(n-2) ENDIF END DEF /* Iterative Method */ DEF fibI(n) a = 0 b = 1 FOR i = 1 TO n temp = a + b a = b b = temp NEXT i fibI = a END DEF /* N-th Term Method */ DEF fibN(n) uphi = .5 + SQR(5)/2 lphi = .5 - SQR(5)/2 fibN = (uphi^n-lphi^n)/SQR(5) END DEF  ## SNOBOL4 ### Recursive snobol define("fib(a)") :(fib_end) fib fib = lt(a,2) a :s(return) fib = fib(a - 1) + fib(a - 2) :(return) fib_end while a = trim(input) :f(end) output = a " " fib(a) :(while) end  ===Tail-recursive=== SNOBOL4 define('trfib(n,a,b)') :(trfib_end) trfib trfib = eq(n,0) a :s(return) trfib = trfib(n - 1, a + b, a) :(return) trfib_end  ### Iterative SNOBOL4 define('ifib(n)f1,f2') :(ifib_end) ifib ifib = le(n,2) 1 :s(return) f1 = 1; f2 = 1 if1 ifib = gt(n,2) f1 + f2 :f(return) f1 = f2; f2 = ifib; n = n - 1 :(if1) ifib_end  ### Analytic {{works with|Macro Spitbol}} {{works with|CSnobol}} Note: Snobol4+ lacks built-in sqrt( ) function. SNOBOL4 define('afib(n)s5') :(afib_end) afib s5 = sqrt(5) afib = (((1 + s5) / 2) ^ n - ((1 - s5) / 2) ^ n) / s5 afib = convert(afib,'integer') :(return) afib_end  Test and display all, Fib 1 .. 10 SNOBOL4 loop i = lt(i,10) i + 1 :f(show) s1 = s1 fib(i) ' ' ; s2 = s2 trfib(i,0,1) ' ' s3 = s3 ifib(i) ' '; s4 = s4 afib(i) ' ' :(loop) show output = s1; output = s2; output = s3; output = s4 end  Output: txt 1 1 2 3 5 8 13 21 34 55 1 1 2 3 5 8 13 21 34 55 1 1 2 3 5 8 13 21 34 55 1 1 2 3 5 8 13 21 34 55  ## SNUSP This is modular SNUSP (which introduces @ and # for threading). ### Iterative snusp @!\+++++++++# /<<+>+>-\ fib\==>>+<!\ ?/\ #</@>\?-<</@>/>+<-\ \-/ \ !\ !\ !\ ?/#  ### Recursive snusp> / ### == \ /> +<<-\ />+<-\ fib==!/?!\-?!\->+>+<>-@\ ### ==?/<@\ ?/<# | #+==/ fib(n-2)|+fib(n-1)| \ ### ==recursion=== /! ### == /  ## Softbridge BASIC ### Iterative basic Function Fibonacci(n) x = 0 y = 1 i = 0 n = ABS(n) If n < 2 Then Fibonacci = n Else Do Until (i = n) sum = x+y x=y y=sum i=i+1 Loop Fibonacci = x End If End Function  ## Spin ### Iterative {{works with|BST/BSTC}} {{works with|FastSpin/FlexSpin}} {{works with|HomeSpun}} {{works with|OpenSpin}} spin con _clkmode = xtal1 + pll16x _clkfreq = 80_000_000 obj ser : "FullDuplexSerial.spin" pub main | i ser.start(31, 30, 0, 115200) repeat i from 0 to 10 ser.dec(fib(i)) ser.tx(32) waitcnt(_clkfreq + cnt) ser.stop cogstop(0) pub fib(i) : b | a b := a := 1 repeat i a := b + (b := a)  {{out}} txt 1 1 2 3 5 8 13 21 34 55 89  ## SPL ### Analytic spl fibo(n)= s5 = #.sqrt(5) <= (((1+s5)/2)^n-((1-s5)/2)^n)/s5 .  ### Iterative spl fibo(n)= ? n<2, <= n f2 = 0 f1 = 1 > i, 2..n f = f1+f2 f2 = f1 f1 = f < <= f .  ### Recursive spl fibo(n)= ? n<2, <= n <= fibo(n-1)+fibo(n-2) .  ## SQL ### Analytic As a running sum: SQL select round ( exp ( sum (ln ( ( 1 + sqrt( 5 ) ) / 2) ) over ( order by level ) ) / sqrt( 5 ) ) fibo from dual connect by level <= 10;  txt FIB ---------- 1 1 2 3 5 8 13 21 34 55 10 rows selected.  As a power: SQL select round ( power( ( 1 + sqrt( 5 ) ) / 2, level ) / sqrt( 5 ) ) fib from dual connect by level <= 10;  txt FIB ---------- 1 1 2 3 5 8 13 21 34 55 10 rows selected.  ### Recursive {{works with|Oracle}} Oracle 12c required sql SQL> with fib(e,f) as (select 1, 1 from dual union all select e+f,e from fib where e <= 55) select f from fib; F ---------- 1 1 2 3 5 8 13 21 34 55 10 rows selected.  {{works with|PostgreSQL}} postgresql CREATE FUNCTION fib(n int) RETURNS numeric AS$$ -- This recursive with generates endless list of Fibonacci numbers. WITH RECURSIVE fibonacci(current, previous) AS ( -- Initialize the current with 0, so the first value will be 0. -- The previous value is set to 1, because its only goal is not -- special casing the zero case, and providing 1 as the second -- number in the sequence. -- -- The numbers end with dots to make them numeric type in -- Postgres. Numeric type has almost arbitrary precision -- (technically just 131,072 digits, but that's good enough for -- most purposes, including calculating huge Fibonacci numbers) SELECT 0., 1. UNION ALL -- To generate Fibonacci number, we need to add together two -- previous Fibonacci numbers. Current number is saved in order -- to be accessed in the next iteration of recursive function. SELECT previous + current, current FROM fibonacci ) -- The user is only interested in current number, not previous. SELECT current FROM fibonacci -- We only need one number, so limit to 1 LIMIT 1 -- Offset the query by the requested argument to get the correct -- position in the list. OFFSET n $$LANGUAGE SQL RETURNS NULL ON NULL INPUT IMMUTABLE;  ## SSEM Calculates the tenth Fibonacci number. To calculate the nth, change the initial value of the counter to n-1 (subject to the restriction that the answer must be small enough to fit in a signed 32-bit integer, the SSEM's only data type). The algorithm is basically straightforward, but the absence of an Add instruction makes the implementation a little more complicated than it would otherwise be. ssem 10101000000000100000000000000000 0. -21 to c acc = -n 01101000000001100000000000000000 1. c to 22 temp = acc 00101000000001010000000000000000 2. Sub. 20 acc -= m 10101000000001100000000000000000 3. c to 21 n = acc 10101000000000100000000000000000 4. -21 to c acc = -n 10101000000001100000000000000000 5. c to 21 n = acc 01101000000000100000000000000000 6. -22 to c acc = -temp 00101000000001100000000000000000 7. c to 20 m = acc 11101000000000100000000000000000 8. -23 to c acc = -count 00011000000001010000000000000000 9. Sub. 24 acc -= -1 00000000000000110000000000000000 10. Test skip next if acc<0 10011000000000000000000000000000 11. 25 to CI goto (15 + 1) 11101000000001100000000000000000 12. c to 23 count = acc 11101000000000100000000000000000 13. -23 to c acc = -count 11101000000001100000000000000000 14. c to 23 count = acc 00011000000000000000000000000000 15. 24 to CI goto (-1 + 1) 10101000000000100000000000000000 16. -21 to c acc = -n 10101000000001100000000000000000 17. c to 21 n = acc 10101000000000100000000000000000 18. -21 to c acc = -n 00000000000001110000000000000000 19. Stop 00000000000000000000000000000000 20. 0 var m = 0 10000000000000000000000000000000 21. 1 var n = 1 00000000000000000000000000000000 22. 0 var temp = 0 10010000000000000000000000000000 23. 9 var count = 9 11111111111111111111111111111111 24. -1 const -1 11110000000000000000000000000000 25. 15 const 15  ## Stata stata program fib args n clear qui set obs n' qui gen a=1 qui replace a=a[_n-1]+a[_n-2] in 3/l end  An implementation using '''[https://www.stata.com/help.cgi?dyngen dyngen]'''. stata program fib args n clear qui set obs n' qui gen a=. dyngen { update a=a[_n-1]+a[_n-2], missval(1) } end fib 10 list  '''Output''' txt +----+ | a | |----| 1. | 1 | 2. | 1 | 3. | 2 | 4. | 3 | 5. | 5 | |----| 6. | 8 | 7. | 13 | 8. | 21 | 9. | 34 | 10. | 55 | +----+  ### Mata stata . mata : function fib(n) { return((((1+sqrt(5))/2):^n-((1-sqrt(5))/2):^n)/sqrt(5)) } : fib(0..10) 1 2 3 4 5 6 7 8 9 10 11 +--------------------------------------------------------+ 1 | 0 1 1 2 3 5 8 13 21 34 55 | +--------------------------------------------------------+ : end  ## StreamIt streamit void->int feedbackloop Fib { join roundrobin(0,1); body in->int filter { work pop 1 push 1 peek 2 { push(peek(0) + peek(1)); pop(); } }; loop Identity; split duplicate; enqueue(0); enqueue(1); }  ## SuperCollider ### Recursive nth fibonacci term for positive n SuperCollider f = { |n| if(n < 2) { n } { f.(n-1) + f.(n-2) } }; (0..20).collect(f)  nth fibonacci term for positive and negative n. SuperCollider f = { |n| var u = neg(sign(n)); if(abs(n) < 2) { n } { f.(2 * u + n) + f.(u + n) } }; (-20..20).collect(f)  ### Analytic SuperCollider ( f = { |n| var sqrt5 = sqrt(5); var p = (1 + sqrt5) / 2; var q = reciprocal(p); ((p ** n) + (q ** n) / sqrt5 + 0.5).trunc }; (0..20).collect(f) )  ### Iterative SuperCollider f = { |n| var a = [1, 1]; n.do { a = a.addFirst(a[0] + a[1]) }; a.reverse }; f.(18)  ## Swift ### Analytic Swift import Cocoa func fibonacci(n: Int) -> Int { let square_root_of_5 = sqrt(5.0) let p = (1 + square_root_of_5) / 2 let q = 1 / p return Int((pow(p,CDouble(n)) + pow(q,CDouble(n))) / square_root_of_5 + 0.5) } for i in 1...30 { println(fibonacci(i)) }  ### Iterative Swift func fibonacci(n: Int) -> Int { if n < 2 { return n } var fibPrev = 1 var fib = 1 for num in 2...n { (fibPrev, fib) = (fib, fib + fibPrev) } return fib }  Sequence: swift func fibonacci() -> SequenceOf { return SequenceOf {() -> GeneratorOf in var window: (UInt, UInt, UInt) = (0, 0, 1) return GeneratorOf { window = (window.1, window.2, window.1 + window.2) return window.0 } } }  ### Recursive Swift func fibonacci(n: Int) -> Int { if n < 2 { return n } else { return fibonacci(n-1) + fibonacci(n-2) } } println(fibonacci(30))  ## Tailspin ### Recursive simple The simplest exponential-time recursive algorithm only handling positive N. Note that the "#" is the tailspin internal recursion which sends the value to the matchers. In this case where there is no initial block and no templates state, we could equivalently write the templates name "nthFibonacci" in place of the "#" to do a normal recursion. tailspin templates nthFibonacci <0|1> ! <> ( - 1 -> #) + ( - 2 -> #) ! end nthFibonacci  ===Iterative, mutable state=== We could use the templates internal mutable state, still only positive N. tailspin templates nthFibonacci @: {n0: 0, n1: 1}; 1.. -> @: {n0: @.n1, n1: @.n0 + @.n1}; @.n0! end nthFibonacci  To handle negatives, we can keep track of the sign and send it to the matchers. tailspin templates nthFibonacci @: {n0: 0, n1: 1}; def sign: -> (<0..> 1! <> -1!); 1..*sign -> sign -> # @.n0! <1> @: {n0: @.n1, n1: @.n0 + @.n1}; <-1> @: {n0: @.n1 - @.n0, n1: @.n0}; end nthFibonacci  ### State machine Instead of mutating state, we could just recurse internally on a state structure. tailspin templates nthFibonacci { N: , n0: 0, n1: 1 } -> # <{ N: <0> }> .n0 ! <{ N: <1..>}> { N: .N - 1, n0: .n1, n1: .n0 + .n1} -> # <> { N: .N + 1, n1: .n0, n0: .n1 - .n0} -> # end nthFibonacci 8 -> nthFibonacci -> '; ' -> !OUT::write -5 -> nthFibonacci -> '; ' -> !OUT::write -6 -> nthFibonacci -> '; ' -> !OUT::write  {{out}} txt 21 5 -8  ## Tcl ### Simple Version These simple versions do not handle negative numbers -- they will return N for N < 2 ### =Iterative= {{trans|Perl}} tcl proc fibiter n { if {n < 2} {return n} set prev 1 set fib 1 for {set i 2} {i < n} {incr i} { lassign [list fib [incr fib prev]] prev fib } return fib }  ### =Recursive= tcl proc fib {n} { if {n < 2} then {expr {n}} else {expr {[fib [expr {n-1}]]+[fib [expr {n-2}]]} } }  The following {{works with|Tcl|8.5}}: defining a procedure in the ::tcl::mathfunc namespace allows that proc to be used as a function in [http://www.tcl.tk/man/tcl8.5/TclCmd/expr.htm expr] expressions. tcl proc tcl::mathfunc::fib {n} { if { n < 2 } { return n } else { return [expr {fib(n-1) + fib(n-2)}] } } # or, more tersely proc tcl::mathfunc::fib {n} {expr {n<2 ? n : fib(n-1) + fib(n-2)}}  E.g.: tcl expr {fib(7)} ;# ==> 13 namespace path tcl::mathfunc #; or, interp alias {} fib {} tcl::mathfunc::fib fib 7 ;# ==> 13  ====Tail-Recursive==== In Tcl 8.6 a ''tailcall'' function is available to permit writing tail-recursive functions in Tcl. This makes deeply recursive functions practical. The availability of large integers also means no truncation of larger numbers. tcl proc fib-tailrec {n} { proc fib:inner {a b n} { if {n < 1} { return a } elseif {n == 1} { return b } else { tailcall fib:inner b [expr {a + b}] [expr {n - 1}] } } return [fib:inner 0 1 n] }  % fib-tailrec 100 354224848179261915075 ### Handling Negative Numbers ### =Iterative= tcl proc fibiter n { if {n < 0} { set n [expr {abs(n)}] set sign [expr {-1**(n+1)}] } else { set sign 1 } if {n < 2} {return n} set prev 1 set fib 1 for {set i 2} {i < n} {incr i} { lassign [list fib [incr fib prev]] prev fib } return [expr {sign * fib}] } fibiter -5 ;# ==> 5 fibiter -6 ;# ==> -8  ### =Recursive= tcl proc tcl::mathfunc::fib {n} {expr {n<-1 ? -1**(n+1) * fib(abs(n)) : n<2 ? n : fib(n-1) + fib(n-2)}} expr {fib(-5)} ;# ==> 5 expr {fib(-6)} ;# ==> -8  ### For the Mathematically Inclined This works up to $fib\left(70\right)$, after which the limited precision of IEEE double precision floating point arithmetic starts to show. {{works with|Tcl|8.5}} tcl proc fib n {expr {round((.5 + .5*sqrt(5)) ** n / sqrt(5))}}  ## Tern ### Recursive tern func fib(n) { if (n < 2) { return 1; } return fib(n - 1) + fib(n - 2); }  ### Coroutine tern func fib(n) { let a = 1; let b = 2; until(n-- <= 0) { yield a; (a, b) = (b, a + b); } }  =={{header|TI-83 BASIC}}== Iterative: ti83b {0,1 While 1 Disp Ans(1 {Ans(2),sum(Ans End  Binet's formula: ti83b Prompt N .5(1+√(5 //golden ratio (Ans^N–(-Ans)^-N)/√(5  =={{header|TI-89 BASIC}}== ### Recursive Optimized implementation (too slow to be usable for ''n'' higher than about 12). ti89b fib(n) when(n<2, n, fib(n-1) + fib(n-2))  ### Iterative Unoptimized implementation (I think the for loop can be eliminated, but I'm not sure). ti89b fib(n) Func Local a,b,c,i 0→a 1→b For i,1,n a→c b→a c+b→b EndFor a EndFunc  ## TSE SAL TSE SAL // library: math: get: series: fibonacci 1.0.0.0.3 (filenamemacro=getmasfi.s) [] [] [kn, ri, su, 20-01-2013 22:04:02] INTEGER PROC FNMathGetSeriesFibonacciI( INTEGER nI ) // // Method: // // 1. Take the sum of the last 2 terms // // 2. Let the sum be the last term // and goto step 1 // INTEGER I = 0 INTEGER minI = 1 INTEGER maxI = nI INTEGER term1I = 0 INTEGER term2I = 1 INTEGER term3I = 0 // FOR I = minI TO maxI // // make value 3 equal to sum of two previous values 1 and 2 // term3I = term1I + term2I // // make value 1 equal to next value 2 // term1I = term2I // // make value 2 equal to next value 3 // term2I = term3I // ENDFOR // RETURN( term3I ) // END PROC Main() STRING s1[255] = "3" REPEAT IF ( NOT ( Ask( " = ", s1, _EDIT_HISTORY_ ) ) AND ( Length( s1 ) > 0 ) ) RETURN() ENDIF Warn( FNMathGetSeriesFibonacciI( Val( s1 ) ) ) // gives e.g. 3 UNTIL FALSE END  ## TUSCRIPT tuscript$$ MODE TUSCRIPT ASK "What fibionacci number do you want?": searchfib="" IF (searchfib!='digits') STOP Loop n=0,{searchfib} IF (n==0) THEN fib=fiba=n ELSEIF (n==1) THEN fib=fibb=n ELSE fib=fiba+fibb, fiba=fibb, fibb=fib ENDIF IF (n!=searchfib) CYCLE PRINT "fibionacci number ",n,"=",fib ENDLOOP  Output: txt What fibionacci number do you want? >12 fibionacci number 12=144  Output: txt What fibionacci number do you want? >31 fibionacci number 31=1346269  Output: txt What fibionacci number do you want? >46 fibionacci 46=1836311903  ## UnixPipes {{incorrect|UnixPipes|There is a race between parallel commands. tee last might open and truncate the file before cat last opens it. Then cat last pipes the empty file to ''xargs'', and ''expr'' reports a syntax error, and the script hangs forever.}} bash echo 1 |tee last fib ; tail -f fib | while read x do cat last | tee -a fib | xargs -n 1 expr$x + |tee last
done


## UNIX Shell

{{works with|bash|3}}

bash
#!/bin/bash

a=0
b=1
max=$1 for (( n=1; "$n" <= "$max";$((n++)) ))
do
a=$(($a + $b)) echo "F($n): $a" b=$(($a -$b))
done


Recursive:
{{works with|bash|3}}

bash
fib() {
local n=$1 [$n -lt 2 ] && echo -n $n || echo -n$(( $( fib$(( n - 1 )) ) + $( fib$(( n - 2 )) ) ))
}


## Ursa

{{trans|Python}}

### Iterative

ursa
def fibIter (int n)
if (< n 2)
return n
end if
decl int fib fibPrev num
set fib (set fibPrev 1)
for (set num 2) (< num n) (inc num)
set fib (+ fib fibPrev)
set fibPrev (- fib fibPrev)
end for
return fib
end


## Ursala

All three methods are shown here, and all have unlimited precision.

Ursala
#import std
#import nat

iterative_fib = ~&/(0,1); ~&r->ll ^|\predecessor ^/~&r sum

recursive_fib = {0,1}^?* iota20


output:

txt

<
<0,0,0>,
<1,1,1>,
<1,1,1>,
<2,2,2>,
<3,3,3>,
<5,5,5>,
<8,8,8>,
<13,13,13>,
<21,21,21>,
<34,34,34>,
<55,55,55>,
<89,89,89>,
<144,144,144>,
<233,233,233>,
<377,377,377>,
<610,610,610>,
<987,987,987>,
<1597,1597,1597>,
<2584,2584,2584>,
<4181,4181,4181>>


## V

Generate n'th fib by using binary recursion

v
[fib
[small?] []
[pred dup pred]
[+]
binrec].


## Vala

### Recursive

Using int, but could easily replace with double, long, ulong, etc.

vala

int fibRec(int n){
if (n < 2)
return n;
else
return fibRec(n - 1) + fibRec(n - 2);
}



### Iterative

Using int, but could easily replace with double, long, ulong, etc.

vala

int fibIter(int n){
if (n < 2)
return n;

int last = 0;
int cur = 1;
int next;

for (int i = 1; i < n; ++i){
next = last + cur;
last = cur;
cur = next;
}

return cur;
}



## VAX Assembly

VAX Assembly
0000  0000     1 .entry	main,0
7E 7CFD  0002     2 	clro	-(sp)			;result buffer
5E   DD  0005     3 	pushl	sp			;pointer to buffer
10   DD  0007     4 	pushl	#16			;descriptor: len of buffer
5B   5E   D0  0009     5 	movl	sp, r11			;-> descriptor
000C     6
7E   01   7D  000C     7 	movq	#1, -(sp)		;init 0,1
000F     8 loop:
7E   6E   04 AE   C1  000F     9 	addl3	4(sp), (sp), -(sp)	;next element on stack
17   1D  0014    10 	bvs	ret			;vs - overflow set, exit
0016    11
5B   DD  0016    12 	pushl	r11			;-> descriptor by ref
04 AE   DF  0018    13 	pushal	4(sp)			;-> fib on stack by ref
00000000'GF   02   FB  001B    14 	calls	#2, g^ots$cvt_l_ti ;convert integer to string 5B DD 0022 15 pushl r11 ; 00000000'GF 01 FB 0024 16 calls #1, g^lib$put_output	;show result
E2   11  002B    17 	brb	loop
002D    18 ret:
04  002D    19 	ret
002E    20 .end	main
$run fib ... 14930352 24157817 39088169 63245986 102334155 165580141 267914296 433494437 701408733 1134903170 1836311903$



## VBA

Like Visual Basic .NET, but with keyword "Public" and type Variant (subtype Currency) instead of Decimal:

vb
Public Function Fib(ByVal n As Integer) As Variant
Dim fib0 As Variant, fib1 As Variant, sum As Variant
Dim i As Integer
fib0 = 0
fib1 = 1
For i = 1 To n
sum = fib0 + fib1
fib0 = fib1
fib1 = sum
Next i
Fib = fib0
End Function


With Currency type, maximum value is fibo(73).

The (slow) recursive version:

VBA

Public Function RFib(Term As Integer) As Long
If Term < 2 Then RFib = Term Else RFib = RFib(Term - 1) + RFib(Term - 2)
End Function



With Long type, maximum value is fibo(46).

## VBScript

===Non-recursive, object oriented, generator===
Defines a generator class, with a default Get property. Uses Currency for larger-than-Long values. Tests for overflow and switches to Double. Overflow information also available from class.

### =Class Definition:=

vb
class generator
dim t1
dim t2
dim tn
dim cur_overflow

Private Sub Class_Initialize
cur_overflow = false
t1 = ccur(0)
t2 = ccur(1)
tn = ccur(t1 + t2)
end sub

public default property get generated
on error resume next

generated = ccur(tn)
if err.number <> 0 then
generated = cdbl(tn)
cur_overflow = true
end if
t1 = ccur(t2)
if err.number <> 0 then
t1 = cdbl(t2)
cur_overflow = true
end if
t2 = ccur(tn)
if err.number <> 0 then
t2 = cdbl(tn)
cur_overflow = true
end if
tn = ccur(t1+ t2)
if err.number <> 0 then
tn = cdbl(t1) + cdbl(t2)
cur_overflow = true
end if
on error goto 0
end property

public property get overflow
overflow = cur_overflow
end property

end class


### =Invocation:=

vb
dim fib
set fib = new generator
dim i
for i = 1 to 100
wscript.stdout.write " " & fib
if fib.overflow then
wscript.echo
exit for
end if
next


### =Output:=

vbscript> 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040 1346269 2178309 3524578 5702887 9227465 14930352 24157817 39088169 63245986 102334155 165580141 267914296 433494437 701408733 1134903170 1836311903 2971215073 4807526976 7778742049 12586269025 20365011074 32951280099 53316291173 86267571272 139583862445 225851433717 365435296162 591286729879 956722026041 1548008755920 2504730781961 4052739537881 6557470319842 10610209857723 17167680177565 27777890035288 44945570212853 72723460248141 117669030460994 190392490709135 308061521170129 498454011879264 806515533049393let memo fib n =
n { > 1 => + (fib (- n 1)) (fib (- n 2)) };


### Iterative

WDTE>let s =
import 'stream';
let a => import 'arrays';

let fib n => (
let reducer p n => [a.at p 1; + (a.at p 0) (a.at p 1)];
s.range 1 n
-> s.reduce [0; 1] reducer
-> a.at 1
;
);


## Whitespace

### Iterative

This program generates Fibonacci numbers until it is [http://ideone.com/VBDLzk forced to terminate].

Whitespace



It was generated from the following pseudo-Assembly.

asm
push 0
push 1

0:
swap
dup
onum
push 10
ochr
copy 1
jump 0


{{out}}

txt
$wspace fib.ws | head -n 6 0 1 1 2 3 5  ### Recursive This program takes a number ''n'' on standard input and outputs the ''n''th member of the Fibonacci sequence. Whitespace  asm ; Read n. push 0 dup inum load ; Call fib(n), ouput the result and a newline, then exit. call 0 onum push 10 ochr exit 0: dup push 2 sub jn 1 ; Return if n < 2. dup push 1 sub call 0 ; Call fib(n - 1). swap ; Get n back into place. push 2 sub call 0 ; Call fib(n - 2). add ; Leave the sum on the stack. 1: ret  {{out}} txt$ echo 10 | wspace fibrec.ws
55


## Wrapl

### Generator

wrapl
DEF fib() (
VAR seq <- [0, 1]; EVERY SUSP seq:values;
REP SUSP seq:put(seq:pop + seq[1])[-1];
);


To get the 17th number:

wrapl
16 SKIP fib();


To get the list of all 17 numbers:

wrapl
ALL 17 OF fib();


### Iterator

Using type match signature to ensure integer argument:

wrapl
TO fib(n @ Integer.T) (
VAR seq <- [0, 1];
EVERY 3:to(n) DO seq:put(seq:pop + seq[1]);
RET seq[-1];
);


## x86 Assembly

{{Works with|MASM}}

asm
TITLE i hate visual studio 4			(Fibs.asm)
;       __         __/--------\
;      >__ \      /  |        |\
;         \  \___/ @  \      /   \__________________
;           \____       \  /                         \\\
;                \____         Coded with love by:    |||
;                      \      Alexander Alvonellos    |||
;                       |          9/29/2011         / ||
;                       |                           |  MM
;                       |      |--------------|     |
;                       |<     |              |<    |
;                       |      |              |     |
;                       |mmmmmm|              |mmmmm|
;; Epic Win.

INCLUDE Irvine32.inc

.data
BEERCOUNT = 48;
Fibs dd 0, 1, BEERCOUNT DUP(0);

.code
main PROC
; I am not responsible for this code.
; They made me write it, against my will.
;Here be dragons
mov esi, offset Fibs; offset array;  ;;were to start (start)
mov ecx, BEERCOUNT; 		;;count of items (how many)
mov ebx, 4; 		;;size (in number of bytes)
call DumpMem;

mov ecx, BEERCOUNT; 	;//http://www.wolframalpha.com/input/?i=F ib%5B47%5D+%3E+4294967295
mov esi, offset Fibs
mov eax, [esi]; 	;//Get me the data from location at ESI
add eax, [esi+4];	;//add into the eax the data at esi + another double (next mem loc)
mov [esi+8], eax;	;//Move that data into the memory location after the second number
add esi, 4;			;//Update the pointer
loop NextPlease;	;//Thank you sir, may I have another?

;Here be dragons
mov esi, offset Fibs; offset array;  ;;were to start (start)
mov ecx, BEERCOUNT; 		;;count of items (how many)
mov ebx, 4; 		;;size (in number of bytes)
call DumpMem;

exit		; exit to operating system
main ENDP

END main


## xEec

This will display the first 93 numbers of the sequence.

xEec

h#1 h#1 h#1 o#
h#10 o$p >f o# h#10 o$ p
ma h? jnext p
t
jnf



## XLISP

### Analytic

Uses Binet's method, based on the golden ratio, which almost feels like cheating—but the task specification doesn't require any particular algorithm, and this one is straightforward and fast.

lisp
(DEFUN FIBONACCI (N)
(FLOOR (+ (/ (EXPT (/ (+ (SQRT 5) 1) 2) N) (SQRT 5)) 0.5)))


To test it, we'll define a RANGE function and ask for the first 50 numbers in the sequence:

lisp
(DEFUN RANGE (X Y)
(IF (<= X Y)
(CONS X (RANGE (+ X 1) Y))))

(PRINT (MAPCAR FIBONACCI (RANGE 1 50)))


{{out}}

txt
(1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040 1346269 2178309 3524578 5702887 9227465 14930352 24157817 39088169 63245986 102334155 165580141 267914296 433494437 701408733 1134903170 1836311903 2971215073 4807526976 7778742049 12586269025)


### Tail recursive

Alternatively, this approach is reasonably efficient:

lisp
(defun fibonacci (x)
(defun fib (a b n)
(if (= n 2)
b
(fib b (+ a b) (- n 1)) ) )
(if (< x 2)
x
(fib 1 1 x) ) )


## Xojo

Pass n to this function where n is the desired number of iterations. This example uses the UInt64 datatype which is as unsigned 64 bit integer. As such, it overflows after the 93rd iteration.

vb
Function fibo(n As Integer) As UInt64

Dim noOne As UInt64 = 1
Dim noTwo As UInt64 = 1
Dim sum As UInt64

For i As Integer = 3 To n
sum = noOne + noTwo
noTwo = noOne
noOne = sum
Next

Return noOne
End Function


## XQuery

xquery
declare function local:fib($n as xs:integer) as xs:integer { if($n < 2)
then $n else local:fib($n - 1) + local:fib(\$n - 2)
};


## zkl

A slight tweak to the task; creates a function that continuously generates fib numbers

zkl
var fibShift=fcn(ab){ab.append(ab.sum()).pop(0)}.fp(L(0,1));


txt

zkl: do(15){ fibShift().print(",") }
0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,

zkl: do(5){ fibShift().print(",") }
610,987,1597,2584,4181,



## ZX Spectrum Basic

### =Iterative=

zxbasic
10 REM Only positive numbers
20 LET n=10
30 LET n1=0: LET n2=1
40 FOR k=1 TO n
50 LET sum=n1+n2
60 LET n1=n2
70 LET n2=sum
80 NEXT k
90 PRINT n1


### =Analytic=

zxbasic
10 DEF FN f(x)=INT (0.5+(((SQR 5+1)/2)^x)/SQR 5)


[[Category:Arithmetic]]