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{{task|Arithmetic operations}} [[Category:Classic CS problems and programs]]
The '''Leonardo numbers''' are a sequence of numbers defined by: L(0) = 1 [1st equation] L(1) = 1 [2nd equation] L(n) = L(n-1) + L(n-2) + 1 [3rd equation] ─── also ─── L(n) = 2 * Fib(n+1) - 1 [4th equation]
:::: where the '''+ 1''' will herein be known as the ''add'' number. :::: where the '''FIB''' is the [[wp:Fibonacci number|Fibonacci number]]s.
The task will be using the 3rd equation (above) to calculate the Leonardo numbers.
[[wp:Edsger W. Dijkstra|Edsger W. Dijkstra]] used them as an integral part of his [[wp:smoothsort|smoothsort]] [[wp:algorithm|algorithm]].
The first few Leonardo numbers are: ''' 1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 ··· '''
;Task: ::* show the 1st '''25''' Leonardo numbers, starting at '''L(0)'''. ::* allow the first two Leonardo numbers to be specified [for '''L(0)''' and '''L(1)''']. ::* allow the ''add'' number to be specified ('''1''' is the default). ::* show the 1st '''25''' Leonardo numbers, specifying '''0''' and '''1''' for '''L(0)''' and '''L(1)''', and '''0''' for the ''add'' number.
(The last task requirement will produce the Fibonacci numbers.)
Show all output here.
;Related tasks:
- [[Fibonacci number]]
- [[Fibonacci n-step number sequences ]]
;See also:
- [[wp:Leonardo number|Wikipedia, Leonardo numbers]]
- [[wp:Fibonacci number|Wikipedia, Fibonacci numbers]]
- [[oeis:A001595|OEIS Leonardo numbers]]
Ada
with Ada.Text_IO; use Ada.Text_IO;
procedure Leonardo is
function Leo
(N : Natural;
Step : Natural := 1;
First : Natural := 1;
Second : Natural := 1) return Natural is
L : array (0..1) of Natural := (First, Second);
begin
for i in 1 .. N loop
L := (L(1), L(0)+L(1)+Step);
end loop;
return L (0);
end Leo;
begin
Put_Line ("First 25 Leonardo numbers:");
for I in 0 .. 24 loop
Put (Integer'Image (Leo (I)));
end loop;
New_Line;
Put_Line ("First 25 Leonardo numbers with L(0) = 0, L(1) = 1, " &
"step = 0 (fibonacci numbers):");
for I in 0 .. 24 loop
Put (Integer'Image (Leo (I, 0, 0, 1)));
end loop;
New_Line;
end Leonardo;
{{out}}
First 25 Leonardo numbers:
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
First 25 Leonardo numbers with L(0) = 0, L(1) = 1, step = 0 (fibonacci numbers):
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
ALGOL 68
BEGIN
# leonardo number parameters #
MODE LEONARDO = STRUCT( INT l0, l1, add number );
# default leonardo number parameters #
LEONARDO leonardo numbers = LEONARDO( 1, 1, 1 );
# operators to allow us to specify non-default parameters #
PRIO WITHLZERO = 9, WITHLONE = 9, WITHADDNUMBER = 9;
OP WITHLZERO = ( LEONARDO parameters, INT l0 )LEONARDO:
LEONARDO( l0, l1 OF parameters, add number OF parameters );
OP WITHLONE = ( LEONARDO parameters, INT l1 )LEONARDO:
LEONARDO( l0 OF parameters, l1, add number OF parameters );
OP WITHADDNUMBER = ( LEONARDO parameters, INT add number )LEONARDO:
LEONARDO( l0 OF parameters, l1 OF parameters, add number );
# show the first n Leonardo numbers with the specified parameters #
PROC show = ( INT n, LEONARDO parameters )VOID:
IF n > 0 THEN
INT l0 = l0 OF parameters;
INT l1 = l1 OF parameters;
INT add number = add number OF parameters;
print( ( whole( l0, 0 ), " " ) );
IF n > 1 THEN
print( ( whole( l1, 0 ), " " ) );
INT lp := l0;
INT ln := l1;
FROM 2 TO n - 1 DO
INT next = ln + lp + add number;
lp := ln;
ln := next;
print( ( whole( ln, 0 ), " " ) )
OD
FI
FI # show # ;
# first series #
print( ( "First 25 Leonardo numbers", newline ) );
show( 25, leonardo numbers );
print( ( newline ) );
# second series #
print( ( "First 25 Leonardo numbers from 0, 1 with add number = 0", newline ) );
show( 25, leonardo numbers WITHLZERO 0 WITHADDNUMBER 0 );
print( ( newline ) )
END
{{out}}
First 25 Leonardo numbers
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
First 25 Leonardo numbers from 0, 1 with add number = 0
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
AppleScript
{{Trans|Python}} (Generator version)
Drawing N items from a non-finite generator:
-- leo :: Int -> Int -> Int -> Generator [Int]
on leo(L0, L1, delta)
script
property x : L0
property y : L1
on |λ|()
set n to x
set {x, y} to {y, x + y + delta}
return n
end |λ|
end script
end leo
-- TEST ---------------------------------------------------
on run
set leonardo to leo(1, 1, 1)
set fibonacci to leo(0, 1, 0)
unlines({"First 25 Leonardo numbers:", ¬
twoLines(take(25, leonardo)), "", ¬
"First 25 Fibonacci numbers:", ¬
twoLines(take(25, fibonacci))})
end run
-- FORMATTING ---------------------------------------------
-- twoLines :: [Int] -> String
on twoLines(xs)
script row
on |λ|(ns)
tab & showList(ns)
end |λ|
end script
return unlines(map(row, chunksOf(16, xs)))
end twoLines
-- GENERIC -----------------------------------------------
-- chunksOf :: Int -> [a] -> [[a]]
on chunksOf(n, xs)
set lng to length of xs
script go
on |λ|(a, i)
set x to (i + n) - 1
if x ≥ lng then
a & {items i thru -1 of xs}
else
a & {items i thru x of xs}
end if
end |λ|
end script
foldl(go, {}, enumFromThenTo(1, n, lng))
end chunksOf
-- enumFromThenTo :: Int -> Int -> Int -> [Int]
on enumFromThenTo(x1, x2, y)
set xs to {}
set d to max(1, (x2 - x1))
repeat with i from x1 to y by d
set end of xs to i
end repeat
return xs
end enumFromThenTo
-- 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
-- intercalate :: String -> [String] -> String
on intercalate(sep, xs)
set {dlm, my text item delimiters} to ¬
{my text item delimiters, sep}
set s to xs as text
set my text item delimiters to dlm
return s
end intercalate
-- Lift 2nd class handler function into 1st class script wrapper
-- mReturn :: First-class m => (a -> b) -> m (a -> b)
on mReturn(f)
if class of f is script then
f
else
script
property |λ| : f
end script
end if
end mReturn
-- map :: (a -> b) -> [a] -> [b]
on map(f, xs)
tell mReturn(f)
set lng to length of xs
set lst to {}
repeat with i from 1 to lng
set end of lst to |λ|(item i of xs, i, xs)
end repeat
return lst
end tell
end map
-- max :: Ord a => a -> a -> a
on max(x, y)
if x > y then
x
else
y
end if
end max
-- showList :: [a] -> String
on showList(xs)
"[" & intercalate(", ", xs) & "]"
end showList
-- take :: Int -> [a] -> [a]
-- take :: Int -> String -> String
on take(n, xs)
set c to class of xs
if list is c then
if 0 < n then
items 1 thru min(n, length of xs) of xs
else
{}
end if
else if string is c then
if 0 < n then
text 1 thru min(n, length of xs) of xs
else
""
end if
else if script is c then
set ys to {}
repeat with i from 1 to n
set v to xs's |λ|()
if missing value is v then
return ys
else
set end of ys to v
end if
end repeat
return ys
else
missing value
end if
end take
-- unlines :: [String] -> String
on unlines(xs)
set {dlm, my text item delimiters} to ¬
{my text item delimiters, linefeed}
set str to xs as text
set my text item delimiters to dlm
str
end unlines
{{Out}}
First 25 Leonardo numbers:
[1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973]
[3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049]
First 25 Fibonacci numbers:
[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]
AWK
# syntax: GAWK -f LEONARDO_NUMBERS.AWK
BEGIN {
leonardo(1,1,1,"Leonardo")
leonardo(0,1,0,"Fibonacci")
exit(0)
}
function leonardo(L0,L1,step,text, i,tmp) {
printf("%s numbers (%d,%d,%d):\n",text,L0,L1,step)
for (i=1; i<=25; i++) {
if (i == 1) {
printf("%d ",L0)
}
else if (i == 2) {
printf("%d ",L1)
}
else {
printf("%d ",L0+L1+step)
tmp = L0
L0 = L1
L1 = tmp + L1 + step
}
}
printf("\n")
}
{{out}}
Leonardo numbers (1,1,1):
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
Fibonacci numbers (0,1,0):
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
BASIC
=
BASIC256
=
subroutine leonardo(L0, L1, suma, texto)
print "Numeros de " + texto + " (" + L0 + "," + L1 + "," + suma + "):"
for i = 1 to 25
if i = 1 then
print L0 + " ";
else
if i = 2 then
print L1 + " ";
else
print L0 + L1 + suma + " ";
tmp = L0
L0 = L1
L1 = tmp + L1 + suma
end if
end if
next i
print chr(10)
end subroutine
#--- Programa Principal ---
call leonardo(1,1,1,"Leonardo")
call leonardo(0,1,0,"Fibonacci")
end
{{out}}
Numeros de Leonardo (1,1,1):
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
Numeros de Fibonacci (0,1,0):
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
==={{header|IS-BASIC}}===
=
## Sinclair ZX81 BASIC
=
Runs on the 1k RAM model with room to spare; hence the long(ish) variable names. The parameters are read from the keyboard.
```basic
10 INPUT L0
20 INPUT L1
30 INPUT ADD
40 PRINT L0;" ";L1;
50 FOR I=3 TO 25
60 LET TEMP=L1
70 LET L1=L0+L1+ADD
80 LET L0=TEMP
90 PRINT " ";L1;
100 NEXT I
{{in}}
1
1
1
{{out}}
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
{{in}}
0
1
0
{{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
BBC BASIC
It's a shame when fonts don't make much of a distinction between l lower-case L and 1 the number One.
leonardo
:
PRINT "Enter values of L0, L1, and ADD, separated by commas:"
INPUT l0%, l1%, add%
PRINT l0% ' l1%
FOR i% = 3 TO 25
temp% = l1%
l1% += l0% + add%
l0% = temp%
PRINT l1%
NEXT
PRINT
END
{{out}}
Enter values of L0, L1, and ADD, separated by commas:
?1, 1, 1
1
1
3
5
9
15
25
41
67
109
177
287
465
753
1219
1973
3193
5167
8361
13529
21891
35421
57313
92735
150049
Enter values of L0, L1, and ADD, separated by commas:
?0, 1, 0
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
Burlesque
blsq ) 1 1 1{.+\/.+}\/+]23!CCLm]wdsh
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
blsq ) 0 1 0{.+\/.+}\/+]23!CCLm]wdsh
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
C
This implementation fulfills the task requirements which state that the first 2 terms and the step increment should be specified. Many other implementations on this page only print out the first 25 numbers.
#include<stdio.h>
void leonardo(int a,int b,int step,int num){
int i,temp;
printf("First 25 Leonardo numbers : \n");
for(i=1;i<=num;i++){
if(i==1)
printf(" %d",a);
else if(i==2)
printf(" %d",b);
else{
printf(" %d",a+b+step);
temp = a;
a = b;
b = temp+b+step;
}
}
}
int main()
{
int a,b,step;
printf("Enter first two Leonardo numbers and increment step : ");
scanf("%d%d%d",&a,&b,&step);
leonardo(a,b,step,25);
return 0;
}
Output : Normal Leonardo Series :
Enter first two Leonardo numbers and increment step : 1 1 1
First 25 Leonardo numbers :
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
Fibonacci Series :
Enter first two Leonardo numbers and increment step : 0 1 0
First 25 Leonardo numbers :
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
C++
#include <iostream>
void leoN( int cnt, int l0 = 1, int l1 = 1, int add = 1 ) {
int t;
for( int i = 0; i < cnt; i++ ) {
std::cout << l0 << " ";
t = l0 + l1 + add; l0 = l1; l1 = t;
}
}
int main( int argc, char* argv[] ) {
std::cout << "Leonardo Numbers: "; leoN( 25 );
std::cout << "\n\nFibonacci Numbers: "; leoN( 25, 0, 1, 0 );
return 0;
}
{{out}}
Leonardo Numbers: 1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
Fibonacci Numbers: 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
C#
{{works with|C sharp|7}}
using System;
using System.Linq;
public class Program
{
public static void Main() {
Console.WriteLine(string.Join(" ", Leonardo().Take(25)));
Console.WriteLine(string.Join(" ", Leonardo(L0: 0, L1: 1, add: 0).Take(25)));
}
public static IEnumerable<int> Leonardo(int L0 = 1, int L1 = 1, int add = 1) {
while (true) {
yield return L0;
(L0, L1) = (L1, L0 + L1 + add);
}
}
}
{{out}}
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
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
Common Lisp
;;;
;;; leo - calculates the first n number from a leo sequence.
;;; The first argument n is the number of values to return. The next three arguments a, b, add are optional.
;;; Default values provide the "original" leonardo numbers as defined in the task.
;;; a and b are the first and second element of the leonardo sequence.
;;; add is the "add number" as defined in the task definition.
;;;
(defun leo (n &optional (a 1) (b 1) (add 1))
(labels ((iterate (n foo)
(if (zerop n) (reverse foo)
(iterate (- n 1)
(cons (+ (first foo) (second foo) add) foo)))))
(cond ((= n 1) (list a))
(T (iterate (- n 2) (list b a))))))
{{out}}
> (leo 25)
(1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049)
> (leo 25 0 1 0)
(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)
Crystal
{{trans|Python}}
def leonardo(l_zero, l_one, add, amount)
terms = [l_zero, l_one]
while terms.size < amount
new = terms[-1] + terms[-2]
new += add
terms << new
end
terms
end
puts "First 25 Leonardo numbers: \n#{ leonardo(1,1,1,25) }"
puts "Leonardo numbers with fibonacci parameters:\n#{ leonardo(0,1,0,25) }"
{{out}}
First 25 Leonardo numbers:
[1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049]
Leonardo numbers with fibonacci parameters:
[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]
D
{{trans|C++}}
import std.stdio;
void main() {
write("Leonardo Numbers: ");
leonardoNumbers( 25 );
write("Fibonacci Numbers: ");
leonardoNumbers( 25, 0, 1, 0 );
}
void leonardoNumbers(int count, int l0=1, int l1=1, int add=1) {
int t;
for (int i=0; i<count; ++i) {
write(l0, " ");
t = l0 + l1 + add;
l0 = l1;
l1 = t;
}
writeln();
}
{{out}}
Leonardo Numbers: 1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
Fibonacci Numbers: 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
=={{header|F#|F sharp}}== {{trans|Haskell}}
open System
let leo l0 l1 d =
Seq.unfold (fun (x, y) -> Some (x, (y, x + y + d))) (l0, l1)
let leonardo = leo 1 1 1
let fibonacci = leo 0 1 0
[<EntryPoint>]
let main _ =
let leoNums = Seq.take 25 leonardo |> Seq.chunkBySize 16
printfn "First 25 of the (1, 1, 1) Leonardo numbers:\n%A" leoNums
Console.WriteLine()
let fibNums = Seq.take 25 fibonacci |> Seq.chunkBySize 16
printfn "First 25 of the (0, 1, 0) Leonardo numbers (= Fibonacci number):\n%A" fibNums
0 // return an integer exit code
{{out}}
First 25 of the (1, 1, 1) Leonardo numbers:
seq
[[|1; 1; 3; 5; 9; 15; 25; 41; 67; 109; 177; 287; 465; 753; 1219; 1973|];
[|3193; 5167; 8361; 13529; 21891; 35421; 57313; 92735; 150049|]]
First 25 of the (0, 1, 0) Leonardo numbers (= Fibonacci number):
seq
[[|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|]]
Factor
: first25-leonardo ( vector add -- seq ) 23 swap '[ dup 2 tail* sum _ + over push ] times ;
: print-leo ( seq -- ) [ pprint bl ] each nl ;
"First 25 Leonardo numbers:" print V{ 1 1 } 1 first25-leonardo print-leo
"First 25 Leonardo numbers with L(0)=0, L(1)=1, add=1:" print V{ 0 1 } 0 first25-leonardo print-leo
{{out}}
```txt
First 25 Leonardo numbers:
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
First 25 Leonardo numbers with L(0)=0, L(1)=1, add=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
Fortran
Happily, no monster values result for the trial run, so ordinary 32-bit integers suffice. The source style uses the F90 facilities only to name the subroutine being ended (i.e. END SUBROUTINE LEONARDO rather than just END) and the I0 format code that shows an integer without a fixed space allowance, convenient in produced well-formed messages. The "$" format code signifies that the end of output from its WRITE statement should not trigger the starting of a new line for the next WRITE statement, convenient when rolling a sequence of values to a line of output one-by-one as they are concocted. Otherwise, the values would have to be accumulated in a suitable array and then written in one go.
Many versions of Fortran have enabled parameters to be optionally supplied and F90 has standardised a protocol, also introducing a declaration syntax that can specify multiple attributes in one statement which in this case would be INTEGER, OPTIONAL:: AF rather than two statements concerning AF. However, in a test run with CALL LEONARDO(25,1,1) the Compaq F90/95 compiler rejected this attempt because there was another invocation with four parameters, not three, in the same program unit. By adding the rigmarole for declaring a MODULE containing the subroutine LEONARDO, its worries would be assuaged. Many compilers (and linkers, for separately-compiled routines) would check neither the number nor the type of parameters so no such complaint would be made - but when run, the code might produce wrong results or crash.
The method relies on producing a sequence of values, rather than calculating L(n) from the start each time a value from the sequence is required.
SUBROUTINE LEONARDO(LAST,L0,L1,AF) !Show the first LAST values of the sequence.
INTEGER LAST !Limit to show.
INTEGER L0,L1 !Starting values.
INTEGER AF !The "Add factor" to deviate from Fibonacci numbers.
OPTIONAL AF !Indicate that this parameter may be omitted.
INTEGER EMBOLISM !The bloat to employ.
INTEGER N,LN,LNL1,LNL2 !Assistants to the calculation.
IF (PRESENT(AF)) THEN !Perhaps the last parameter has not been given.
EMBOLISM = AF !It has. Take its value.
ELSE !But if not,
EMBOLISM = 1 !This is the specified default.
END IF !Perhaps there should be some report on this?
WRITE (6,1) LAST,L0,L1,EMBOLISM !Announce.
1 FORMAT ("The first ",I0, !The I0 format code avoids excessive spacing.
1 " numbers in the Leonardo sequence defined by L(0) = ",I0,
2 " and L(1) = ",I0," with L(n) = L(n - 1) + L(n - 2) + ",I0)
IF (LAST .GE. 1) WRITE (6,2) L0 !In principle, LAST may be small.
IF (LAST .GE. 2) WRITE (6,2) L1 !!So, suspicion rules.
2 FORMAT (I0,", ",$) !Obviously, the $ sez "don't finish the line".
LNL1 = L0 !Syncopation for the sequence's initial values.
LN = L1 !Since the parameters ought not be damaged.
DO N = 3,LAST !Step away.
LNL2 = LNL1 !Advance the two state variables one step.
LNL1 = LN !Ready to make a step forward.
LN = LNL1 + LNL2 + EMBOLISM !Thus.
WRITE (6,2) LN !Reveal the value. Overflow is distant...
END DO !On to the next step.
WRITE (6,*) !Finish the line.
END SUBROUTINE LEONARDO !Only speedy for the sequential production of values.
PROGRAM POKE
CALL LEONARDO(25,1,1,1) !The first 25 Leonardo numbers.
CALL LEONARDO(25,0,1,0) !Deviates to give the Fibonacci sequence.
END
Output:
The first 25 numbers in the Leonardo sequence defined by L(0) = 1 and L(1) = 1 with L(n) = L(n - 1) + L(n - 2) + 1
1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049,
The first 25 numbers in the Leonardo sequence defined by L(0) = 0 and L(1) = 1 with L(n) = L(n - 1) + L(n - 2) + 0
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,
FreeBASIC
Sub leonardo(L0 As Integer, L1 As Integer, suma As Integer, texto As String)
Dim As Integer i, tmp
Print "Numeros de " &texto &" (" &L0 &"," &L1 &"," &suma &"):"
For i = 1 To 25
If i = 1 Then
Print L0;
Elseif i = 2 Then
Print L1;
Else
Print L0 + L1 + suma;
tmp = L0
L0 = L1
L1 = tmp + L1 + suma
End If
Next i
Print Chr(10)
End Sub
'--- Programa Principal ---
leonardo(1,1,1,"Leonardo")
leonardo(0,1,0,"Fibonacci")
End
{{out}}
Numeros de Leonardo (1,1,1):
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
Numeros de Fibonacci (0,1,0):
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
Go
package main
import "fmt"
func leonardo(n, l0, l1, add int) []int {
leo := make([]int, n)
leo[0] = l0
leo[1] = l1
for i := 2; i < n; i++ {
leo[i] = leo[i - 1] + leo[i - 2] + add
}
return leo
}
func main() {
fmt.Println("The first 25 Leonardo numbers with L[0] = 1, L[1] = 1 and add number = 1 are:")
fmt.Println(leonardo(25, 1, 1, 1))
fmt.Println("\nThe first 25 Leonardo numbers with L[0] = 0, L[1] = 1 and add number = 0 are:")
fmt.Println(leonardo(25, 0, 1, 0))
}
{{out}}
The first 25 Leonardo numbers with L[0] = 1, L[1] = 1 and add number = 1 are:
[1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049]
The first 25 Leonardo numbers with L[0] = 0, L[1] = 1 and add number = 0 are:
[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]
Haskell
import Data.List.Split (chunksOf)
import Data.List (unfoldr)
-- LEONARDO NUMBERS -----------------------------------------------------------
-- L0 -> L1 -> Add number -> Series (infinite)
leo :: Integer -> Integer -> Integer -> [Integer]
leo l0 l1 d = unfoldr (\(x, y) -> Just (x, (y, x + y + d))) (l0, l1)
leonardo :: [Integer]
leonardo = leo 1 1 1
fibonacci :: [Integer]
fibonacci = leo 0 1 0
-- TEST -----------------------------------------------------------------------
main :: IO ()
main = do
let twoLines = unlines . fmap (('\t' :) . show) . chunksOf 16
(putStrLn . unlines)
[ "First 25 default (1, 1, 1) Leonardo numbers:\n"
, twoLines $ take 25 leonardo
, "First 25 of the (0, 1, 0) Leonardo numbers (= Fibonacci numbers):\n"
, twoLines $ take 25 fibonacci
]
{{Out}}
First 25 default (1, 1, 1) Leonardo numbers:
[1,1,3,5,9,15,25,41,67,109,177,287,465,753,1219,1973]
[3193,5167,8361,13529,21891,35421,57313,92735,150049]
First 25 of the (0, 1, 0) Leonardo numbers (= Fibonacci numbers):
[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]
J
leo =: (] , {.@[ + _2&{@] + {:@])^:(_2&+@{:@[)
{{Out}}
1 25 leo 1 1
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
0 25 leo 0 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
Java
{{trans|Kotlin}}
import java.util.Arrays;
import java.util.List;
@SuppressWarnings("SameParameterValue")
public class LeonardoNumbers {
private static List<Integer> leonardo(int n) {
return leonardo(n, 1, 1, 1);
}
private static List<Integer> leonardo(int n, int l0, int l1, int add) {
Integer[] leo = new Integer[n];
leo[0] = l0;
leo[1] = l1;
for (int i = 2; i < n; i++) {
leo[i] = leo[i - 1] + leo[i - 2] + add;
}
return Arrays.asList(leo);
}
public static void main(String[] args) {
System.out.println("The first 25 Leonardo numbers with L[0] = 1, L[1] = 1 and add number = 1 are:");
System.out.println(leonardo(25));
System.out.println("\nThe first 25 Leonardo numbers with L[0] = 0, L[1] = 1 and add number = 0 are:");
System.out.println(leonardo(25, 0, 1, 0));
}
}
{{out}}
The first 25 Leonardo numbers with L[0] = 1, L[1] = 1 and add number = 1 are:
[1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049]
The first 25 Leonardo numbers with L[0] = 0, L[1] = 1 and add number = 0 are:
[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]
JavaScript
ES6
const leoNum = (c, l0=1, l1=1, add=1) => new Array(c).fill(add).reduce(
(p, c, i) => i > 1 ? p.push(p[i-1] + p[i-2] + c) && p : p, [l0, l1]
);
console.log(leoNum(25));
console.log(leoNum(25, 0, 1, 0));
[1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049]
[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]
Or, taking N terms from a non-finite Javascript generator: {{Trans|Python}}
(() => {
'use strict';
// leo :: Int -> Int -> Int -> Generator [Int]
function* leo(L0, L1, delta) {
let [x, y] = [L0, L1];
while (true) {
yield x;
[x, y] = [y, x + y + delta];
}
}
// main :: IO ()
const main = () => {
const
leonardo = leo(1, 1, 1),
fibonacci = leo(0, 1, 0);
console.log(
unlines([
'First 25 Leonardo numbers:',
twoLines(take(25, leonardo)),
'',
'First 25 Fibonacci numbers:',
twoLines(take(25, fibonacci))
])
);
};
// FORMATTING -----------------------------------------
// twoLines :: [Int] -> String
const twoLines = xs =>
unlines(map(
ns => '\t' + showJSON(ns),
chunksOf(16, xs)
));
// GENERIC FUNCTIONS ----------------------------------
// chunksOf :: Int -> [a] -> [[a]]
const chunksOf = (n, xs) =>
enumFromThenTo(0, n - 1, xs.length - 1)
.reduce(
(a, i) => a.concat([xs.slice(i, i + n)]),
[]
);
// enumFromThenTo :: Int -> Int -> Int -> [Int]
const enumFromThenTo = (x1, x2, y) => {
const d = x2 - x1;
return Array.from({
length: Math.floor(y - x2) / d + 2
}, (_, i) => x1 + (d * i));
};
// map :: (a -> b) -> [a] -> [b]
const map = (f, xs) => xs.map(f);
// showJSON :: a -> String
const showJSON = x => JSON.stringify(x);
// take :: Int -> [a] -> [a]
// take :: Int -> String -> String
const take = (n, xs) =>
'GeneratorFunction' !== xs.constructor.constructor.name ? (
xs.slice(0, n)
) : [].concat.apply([], Array.from({
length: n
}, () => {
const x = xs.next();
return x.done ? [] : [x.value];
}));
// unlines :: [String] -> String
const unlines = xs => xs.join('\n');
// MAIN ---
return main();
})();
{{Out}}
First 25 Leonardo numbers:
[1,1,3,5,9,15,25,41,67,109,177,287,465,753,1219,1973]
[3193,5167,8361,13529,21891,35421,57313,92735,150049]
First 25 Fibonacci numbers:
[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]
jq
Naive Implementation
def Leonardo(zero; one; incr):
def leo:
if . == 0 then zero
elif . == 1 then one
else ((.-1) |leo) + ((.-2) | leo) + incr
end;
leo;
Implementation with Caching
An array is used for caching, with .[n] storing the value L(n).
def Leonardo(zero; one; incr):
def leo(n):
if .[n] then .
else leo(n-1) # optimization of leo(n-2)|leo(n-1)
| .[n] = .[n-1] + .[n-2] + incr
end;
. as $n | [zero,one] | leo($n) | .[$n];
(To compute the sequence of Leonardo numbers L(1) ... L(n) without redundant computation, the last element of the pipeline in the last line of the function above should be dropped.)
'''Examples'''
[range(0;25) | Leonardo(1;1;1)]
{{out}}
[1,1,3,5,9,15,25,41,67,109,177,287,465,753,1219,1973,3193,5167,8361,13529,21891,35421,57313,92735,150049]
[range(0;25) | Leonardo(0;1;0)]
{{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]
Julia
{{works with|Julia|0.6}}
function L(n, add::Int=1, firsts::Vector=[1, 1])
l = max(maximum(n) .+ 1, length(firsts))
r = Vector{Int}(l)
r[1:length(firsts)] = firsts
for i in 3:l
r[i] = r[i - 1] + r[i - 2] + add
end
return r[n .+ 1]
end
# Task 1
println("First 25 Leonardo numbers: ", join(L(0:24), ", "))
# Task 2
@show L(0) L(1)
# Task 4
println("First 25 Leonardo numbers starting with [0, 1]: ", join(L(0:24, 0, [0, 1]), ", "))
{{out}}
First 25 Leonardo numbers: 1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049
L(0) = 1
L(1) = 1
First 25 Leonardo numbers starting with 0, 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
Kotlin
// version 1.1.2
fun leonardo(n: Int, l0: Int = 1, l1: Int = 1, add: Int = 1): IntArray {
val leo = IntArray(n)
leo[0] = l0
leo[1] = l1
for (i in 2 until n) leo[i] = leo[i - 1] + leo[i - 2] + add
return leo
}
fun main(args: Array<String>) {
println("The first 25 Leonardo numbers with L[0] = 1, L[1] = 1 and add number = 1 are:")
println(leonardo(25).joinToString(" "))
println("\nThe first 25 Leonardo numbers with L[0] = 0, L[1] = 1 and add number = 0 are:")
println(leonardo(25, 0, 1, 0).joinToString(" "))
}
{{out}}
The first 25 Leonardo numbers with L[0] = 1, L[1] = 1 and add number = 1 are:
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
The first 25 Leonardo numbers with L[0] = 0, L[1] = 1 and add number = 0 are:
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
Lua
function leoNums (n, L0, L1, add)
local L0, L1, add = L0 or 1, L1 or 1, add or 1
local lNums, nextNum = {L0, L1}
while #lNums < n do
nextNum = lNums[#lNums] + lNums[#lNums - 1] + add
table.insert(lNums, nextNum)
end
return lNums
end
function show (msg, t)
print(msg .. ":")
for i, x in ipairs(t) do
io.write(x .. " ")
end
print("\n")
end
show("Leonardo numbers", leoNums(25))
show("Fibonacci numbers", leoNums(25, 0, 1, 0))
{{out}}
Leonardo numbers:
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
Fibonacci numbers:
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
=={{Header|Maple}}==
L := proc(n, L_0, L_1, add)
if n = 0 then
return L_0;
elif n = 1 then
return L_1;
else
return L(n - 1) + L(n - 2) + add;
end if;
end proc:
Leonardo := n -> (L(1, 1, 1),[seq(0..n - 1)])
Fibonacci := n -> (L(0, 1, 0), [seq(0..n - 1)])
[1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049]
[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]
=={{Header|Mathematica}}==
{{incorrect|Mathematica|
The wrong formula is being used (the 4th formula is being used, instead, the 3rd formula is to be used.
Also, output is missing for the Fibonacci series calculated via the Leonardo series, the 3rd formula.
}}
L[n_] := 2 Fibonacci[n + 1] - 1; L /@ Range[25]
{1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049, 242785}
min
{{works with|min|0.19.3}}
(over over + rolldown pop pick +) :next
(('print dip " " print! next) 25 times newline) :leo
"First 25 Leonardo numbers:" puts!
1 1 1 leo
"First 25 Leonardo numbers with add=0, L(0)=0, L(1)=1:" puts!
0 0 1 leo
{{out}}
First 25 Leonardo numbers:
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
First 25 Leonardo numbers with add=0, L(0)=0, L(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
=={{header|Modula-2}}==
MODULE Leonardo;
FROM FormatString IMPORT FormatString;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
PROCEDURE leonardo(a,b,step,num : INTEGER);
VAR
buf : ARRAY[0..63] OF CHAR;
i,temp : INTEGER;
BEGIN
FOR i:=1 TO num DO
IF i=1 THEN
FormatString(" %i", buf, a);
WriteString(buf)
ELSIF i=2 THEN
FormatString(" %i", buf, b);
WriteString(buf)
ELSE
FormatString(" %i", buf, a+b+step);
WriteString(buf);
temp := a;
a := b;
b := temp + b + step
END
END;
WriteLn
END leonardo;
BEGIN
leonardo(1,1,1,25);
leonardo(0,1,0,25);
ReadChar
END Leonardo.
Nim
import strformat
proc leonardoNumbers(count: int, L0: int = 1,
L1: int = 1, ADD: int = 1) =
var t = 0
var (L0_loc, L1_loc) = (L0, L1)
for i in 0..<count:
write(stdout, fmt"{L0_loc:7}")
t = L0_loc + L1_loc + ADD
L0_loc = L1_loc
L1_loc = t
if i mod 5 == 4:
write(stdout, "\n")
write(stdout, "\n")
echo "Leonardo Numbers:"
leonardoNumbers(25)
echo "Fibonacci Numbers: "
leonardoNumbers(25, 0, 1, 0)
{{out}}
Leonardo Numbers:
1 1 3 5 9
15 25 41 67 109
177 287 465 753 1219
1973 3193 5167 8361 13529
21891 35421 57313 92735 150049
Fibonacci Numbers:
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
Perl
no warnings 'experimental::signatures';
use feature 'signatures';
sub leonardo ($n, $l0 = 1, $l1 = 1, $add = 1) {
($l0, $l1) = ($l1, $l0+$l1+$add) for 1..$n;
$l0;
}
my @L = map { leonardo($_) } 0..24;
print "Leonardo[1,1,1]: @L\n";
my @F = map { leonardo($_,0,1,0) } 0..24;
print "Leonardo[0,1,0]: @F\n";
{{out}}
Leonardo[1,1,1]: 1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
Leonardo[0,1,0]: 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
Perl 6
sub 𝑳 ( $𝑳0 = 1, $𝑳1 = 1, $𝑳add = 1 ) { $𝑳0, $𝑳1, { $^n2 + $^n1 + $𝑳add } ... * }
# Part 1
say "The first 25 Leonardo numbers:";
put 𝑳()[^25];
# Part 2
say "\nThe first 25 numbers using 𝑳0 of 0, 𝑳1 of 1, and adder of 0:";
put 𝑳( 0, 1, 0 )[^25];
{{out}}
The first 25 Leonardo numbers:
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
The first 25 numbers using 𝑳0 of 0, 𝑳1 of 1, and adder of 0:
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368
Phix
function leonardo(integer n, l1=1, l2=1, step=1)
--return the first n leonardo numbers, starting {l1,l2}, with step as the add number
sequence res = {l1,l2}
while length(res)<n do
res = append(res,res[$]+res[$-1]+step)
end while
return res
end function
?{"Leonardo",leonardo(25)}
?{"Fibonacci",leonardo(25,0,1,0)}
{{out}}
{"Leonardo",{1,1,3,5,9,15,25,41,67,109,177,287,465,753,1219,1973,3193,5167,8361,13529,21891,35421,57313,92735,150049}}
{"Fibonacci",{0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,17711,28657,46368}}
PicoLisp
(de leo (A B C)
(default A 1 B 1 C 1)
(make
(do 25
(inc
'B
(+ (link (swap 'A B)) C) ) ) ) )
(println 'Leonardo (leo))
(println 'Fibonacci (leo 0 1 0))
{{out}}
Leonardo (1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049)
Fibonacci (0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368)
PureBasic
EnableExplicit
#N = 25
Procedure leon_R(a.i, b.i, s.i = 1, n.i = #N)
If n>2
Print(Space(1) + Str(a + b + s))
ProcedureReturn leon_R(b, a + b + s, s, n-1)
EndIf
EndProcedure
If OpenConsole()
Define r$
Print("Enter first two Leonardo numbers and increment step (separated by space) : ")
r$ = Input()
PrintN("First " + Str(#N) + " Leonardo numbers : ")
Print(StringField(r$, 1, Chr(32)) + Space(1) +
StringField(r$, 2, Chr(32)))
leon_R(Val(StringField(r$, 1, Chr(32))),
Val(StringField(r$, 2, Chr(32))),
Val(StringField(r$, 3, Chr(32))))
r$ = Input()
EndIf
{{out}}
Enter first two Leonardo numbers and increment step (separated by space) : 1 1 1
First 25 Leonardo numbers :
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
Enter first two Leonardo numbers and increment step (separated by space) : 0 1 0
First 25 Leonardo numbers :
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
Python
Finite iteration
def Leonardo(L_Zero, L_One, Add, Amount):
terms = [L_Zero,L_One]
while len(terms) < Amount:
new = terms[-1] + terms[-2]
new += Add
terms.append(new)
return terms
out = ""
print "First 25 Leonardo numbers:"
for term in Leonardo(1,1,1,25):
out += str(term) + " "
print out
out = ""
print "Leonardo numbers with fibonacci parameters:"
for term in Leonardo(0,1,0,25):
out += str(term) + " "
print out
{{out}}
First 25 Leonardo numbers:
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
Leonardo numbers with fibonacci parameters:
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
===Non-finite generation=== Or, for a non-finite stream of Leonardos, we can use a Python generator: {{Works with|Python|3}}
'''Leonardo numbers'''
from functools import (reduce)
from itertools import (islice)
# leo :: Int -> Int -> Int -> Generator [Int]
def leo(L0, L1, delta):
'''A number series of the
Leonardo and Fibonacci pattern,
where L0 and L1 are the first two terms,
and delta = 1 for (L0, L1) == (1, 1)
yields the Leonardo series, while
delta = 0 defines the Fibonacci series.'''
(x, y) = (L0, L1)
while True:
yield x
(x, y) = (y, x + y + delta)
# main :: IO()
def main():
'''Tests.'''
print('\n'.join([
'First 25 Leonardo numbers:',
folded(16)(take(25)(
leo(1, 1, 1)
)),
'',
'First 25 Fibonacci numbers:',
folded(16)(take(25)(
leo(0, 1, 0)
))
]))
# FORMATTING ----------------------------------------------
# folded :: Int -> [a] -> String
def folded(n):
'''Long list folded to rows of n terms each.'''
return lambda xs: '[' + ('\n '.join(
str(ns)[1:-1] for ns in chunksOf(n)(xs)
) + ']')
# GENERIC -------------------------------------------------
# chunksOf :: Int -> [a] -> [[a]]
def chunksOf(n):
'''A series of lists of length n,
subdividing the contents of xs.
Where the length of xs is not evenly divible,
the final list will be shorter than n.'''
return lambda xs: reduce(
lambda a, i: a + [xs[i:n + i]],
range(0, len(xs), n), []
) if 0 < n else []
# take :: Int -> [a] -> [a]
# take :: Int -> String -> String
def take(n):
'''The prefix of xs of length n,
or xs itself if n > length xs.'''
return lambda xs: (
xs[0:n]
if isinstance(xs, list)
else list(islice(xs, n))
)
# MAIN ---
if __name__ == '__main__':
main()
{{Out}}
First 25 Leonardo numbers:
[1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973
3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049]
First 25 Fibonacci numbers:
[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]
R
leonardo_numbers <- function(add = 1, l0 = 1, l1 = 1, how_many = 25) {
result <- c(l0, l1)
for (i in 3:how_many)
result <- append(result, result[[i - 1]] + result[[i - 2]] + add)
result
}
cat("First 25 Leonardo numbers\n")
cat(leonardo_numbers(), "\n")
cat("First 25 Leonardo numbers from 0, 1 with add number = 0\n")
cat(leonardo_numbers(0, 0, 1), "\n")
{{out}}
First 25 Leonardo numbers
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
First 25 Leonardo numbers from 0, 1 with add number = 0
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
Racket
#lang racket
(define (Leonardo n #:L0 (L0 1) #:L1 (L1 1) #:1+ (1+ 1))
(cond [(= n 0) L0]
[(= n 1) L1]
[else
(let inr ((n (- n 2)) (L_n-2 L0) (L_n-1 L1))
(let ((L_n (+ L_n-1 L_n-2 1+)))
(if (zero? n) L_n (inr (sub1 n) L_n-1 L_n))))]))
(module+ main
(map Leonardo (range 25))
(map (curry Leonardo #:L0 0 #:L1 1 #:1+ 0) (range 25)))
(module+ test
(require rackunit)
(check-equal? (Leonardo 0) 1)
(check-equal? (Leonardo 1) 1)
(check-equal? (Leonardo 2) 3)
(check-equal? (Leonardo 3) 5))
{{out}}
'(1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049)
'(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)
REXX
/*REXX pgm computes Leonardo numbers, allowing the specification of L(0), L(1), and ADD#*/
numeric digits 500 /*just in case the user gets ka-razy. */
@.=1 /*define the default for the @. array.*/
parse arg N L0 L1 a# . /*obtain optional arguments from the CL*/
if N =='' | N =="," then N= 25 /*Not specified? Then use the default.*/
if L0\=='' & L0\=="," then @.0= L0 /*Was " " " " value. */
if L1\=='' & L1\=="," then @.1= L1 /* " " " " " " */
if a#\=='' & a#\=="," then @.a= a# /* " " " " " " */
say 'The first ' N " Leonardo numbers are:" /*display a title for the output series*/
if @.0\==1 | @.1\==1 then say 'using ' @.0 " for L(0)"
if @.0\==1 | @.1\==1 then say 'using ' @.1 " for L(1)"
if @.a\==1 then say 'using ' @.a " for the add number"
say /*display blank line before the output.*/
$= /*initialize the output line to "null".*/
do j=0 for N /*construct a list of Leonardo numbers.*/
if j<2 then z=@.j /*for the 1st two numbers, use the fiat*/
else do /*··· otherwise, compute the Leonardo #*/
_=@.0 /*save the old primary Leonardo number.*/
@.0=@.1 /*store the new primary number in old. */
@.1=@.0 + _ + @.a /*compute the next Leonardo number. */
z=@.1 /*store the next Leonardo number in Z. */
end /* [↑] only 2 Leonardo #s are stored. */
$=$ z /*append the just computed # to $ list.*/
end /*j*/ /* [↓] elide the leading blank in $. */
say strip($) /*stick a fork in it, we're all done. */
{{out|output|text= when using the default input:}}
The first 25 Leonardo numbers are:
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
{{out|output|text= when using the input of: 12 0 1 0 }}
The first 25 Leonardo numbers are:
using 0 for L(0)
using 1 for L(1)
using 0 for the add number
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
Ring
# Project : Leanardo numbers
n0 = 1
n1 = 1
add = 1
see "First 25 Leonardo numbers:" + nl
leonardo()
n0 = 1
n1 = 1
add = 0
see "First 25 Leonardo numbers with L(0) = 0, L(1) = 1, step = 0 (fibonacci numbers):" + nl
see "" + add + " "
leonardo()
func leonardo()
see "" + n0 + " " + n1
for i=3 to 25
temp=n1
n1=n0+n1+add
n0=temp
see " "+ n1
next
see nl
Output:
First 25 Leonardo numbers:
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
First 25 Leonardo numbers with L(0) = 0, L(1) = 1, step = 0 (fibonacci numbers):
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
Ruby
Enumerators are nice for this.
def leonardo(l0=1, l1=1, add=1)
return to_enum(__method__,l0,l1,add) unless block_given?
loop do
yield l0
l0, l1 = l1, l0+l1+add
end
end
p leonardo.take(25)
p leonardo(0,1,0).take(25)
{{out}}
[1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049]
[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]
Run BASIC
sqliteconnect #mem, ":memory:"
#mem execute("CREATE TABLE lno (name,L0,L1,ad)")
#mem execute("INSERT INTO lno VALUES('Leonardo',1,1,1),('Fibonacci',0,1,0);")
#mem execute("SELECT * FROM lno")
for j = 1 to 2
#row = #mem #nextrow()
name$ = #row name$()
L0 = #row L0()
L1 = #row L1()
ad = #row ad()
print :print name$;" add=";ad :print" ";L0;" ";L1;" ";
for i = 3 to 25
temp = L1
L1 = L0 + L1 + ad
L0 = temp
print L1;" ";
next i
next j
end
Leonardo add=1
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
Fibonacci add=0
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
Scala
def leo( n:Int, n1:Int=1, n2:Int=1, addnum:Int=1 ) : BigInt = n match {
case 0 => n1
case 1 => n2
case n => leo(n - 1, n1, n2, addnum) + leo(n - 2, n1, n2, addnum) + addnum
}
{
println( "The first 25 Leonardo Numbers:")
(0 until 25) foreach { n => print( leo(n) + " " ) }
println( "\n\nThe first 25 Fibonacci Numbers:")
(0 until 25) foreach { n => print( leo(n, n1=0, n2=1, addnum=0) + " " ) }
}
{{out}}
The first 25 Leonardo Numbers:
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
The first 25 Fibonacci Numbers:
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
Seed7
$ include "seed7_05.s7i";
const proc: leonardo (in var integer: l0, in var integer: l1, in integer: add, in integer: count) is func
local
var integer: temp is 0;
begin
for count do
write(" " <& l0);
temp := l0 + l1 + add;
l0 := l1;
l1 := temp;
end for;
writeln;
end func;
const proc: main is func
begin
write("Leonardo Numbers:");
leonardo(1, 1, 1, 25);
write("Fibonacci Numbers:");
leonardo(0, 1, 0, 25);
end func;
{{out}}
Leonardo Numbers: 1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
Fibonacci Numbers: 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
Sidef
func 𝑳(n, 𝑳0 = 1, 𝑳1 = 1, 𝑳add = 1) {
{ (𝑳0, 𝑳1) = (𝑳1, 𝑳0 + 𝑳1 + 𝑳add) } * n
return 𝑳0
}
say "The first 25 Leonardo numbers:"
say 25.of { 𝑳(_) }
say "\nThe first 25 numbers using 𝑳0 of 0, 𝑳1 of 1, and adder of 0:"
say 25.of { 𝑳(_, 0, 1, 0) }
{{out}}
The first 25 Leonardo numbers:
[1, 1, 3, 5, 9, 15, 25, 41, 67, 109, 177, 287, 465, 753, 1219, 1973, 3193, 5167, 8361, 13529, 21891, 35421, 57313, 92735, 150049]
The first 25 numbers using 𝑳0 of 0, 𝑳1 of 1, and adder of 0:
[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]
VBA
Option Explicit
Private Sub LeonardoNumbers()
Dim L, MyString As String
Debug.Print "First 25 Leonardo numbers :"
L = Leo_Numbers(25, 1, 1, 1)
MyString = Join(L, "; ")
Debug.Print MyString
Debug.Print "First 25 Leonardo numbers from 0, 1 with add number = 0"
L = Leo_Numbers(25, 0, 1, 0)
MyString = Join(L, "; ")
Debug.Print MyString
Debug.Print "If the first prarameter is too small :"
L = Leo_Numbers(1, 0, 1, 0)
MyString = Join(L, "; ")
Debug.Print MyString
End Sub
Public Function Leo_Numbers(HowMany As Long, L_0 As Long, L_1 As Long, Add_Nb As Long)
Dim N As Long, Ltemp
If HowMany > 1 Then
ReDim Ltemp(HowMany - 1)
Ltemp(0) = L_0: Ltemp(1) = L_1
For N = 2 To HowMany - 1
Ltemp(N) = Ltemp(N - 1) + Ltemp(N - 2) + Add_Nb
Next N
Else
ReDim Ltemp(0)
Ltemp(0) = "The first parameter is too small"
End If
Leo_Numbers = Ltemp
End Function
{{out}}
First 25 Leonardo numbers :
1; 1; 3; 5; 9; 15; 25; 41; 67; 109; 177; 287; 465; 753; 1219; 1973; 3193; 5167; 8361; 13529; 21891; 35421; 57313; 92735; 150049
First 25 Leonardo numbers from 0, 1 with add number = 0
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
If the first prarameter is too small :
The first parameter is too small
Visual Basic .NET
{{trans|C#}}
Module Module1
Iterator Function Leonardo(Optional L0 = 1, Optional L1 = 1, Optional add = 1) As IEnumerable(Of Integer)
While True
Yield L0
Dim t = L0 + L1 + add
L0 = L1
L1 = t
End While
End Function
Sub Main()
Console.WriteLine(String.Join(" ", Leonardo().Take(25)))
Console.WriteLine(String.Join(" ", Leonardo(0, 1, 0).Take(25)))
End Sub
End Module
{{out}}
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
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
zkl
fcn leonardoNumber(n, n1=1,n2=1,addnum=1){
if(n==0) return(n1);
if(n==1) return(n2);
self.fcn(n-1,n1,n2,addnum) + self.fcn(n-2,n1,n2,addnum) + addnum
}
println("The first 25 Leonardo Numbers:");
foreach n in (25){ print(leonardoNumber(n)," ") }
println("\n");
println("The first 25 Fibonacci Numbers:");
foreach n in (25){ print(leonardoNumber(n, 0,1,0)," ") }
println();
{{out}}
The first 25 Leonardo Numbers:
1 1 3 5 9 15 25 41 67 109 177 287 465 753 1219 1973 3193 5167 8361 13529 21891 35421 57313 92735 150049
The first 25 Fibonacci Numbers:
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