⚠️ Warning: This is a draft ⚠️

This means it might contain formatting issues, incorrect code, conceptual problems, or other severe issues.

If you want to help to improve and eventually enable this page, please fork RosettaGit's repository and open a merge request on GitHub.

;Definitions: The '''fusc''' integer sequence is defined as: ::* fusc(0) = 0 ::* fusc(1) = 1 ::* for '''n'''>1, the '''n'''th term is defined as: ::::* if '''n''' is even; fusc(n) = fusc(n/2) ::::* if '''n''' is odd; fusc(n) = fusc((n-1)/2) + fusc((n+1)/2)

Note that MathWorld's definition starts with unity, not zero. This task will be using the OEIS' version (above).

;An observation: :::::* fusc(A) = fusc(B)

where '''A''' is some non-negative integer expressed in binary, and where '''B''' is the binary value of '''A''' reversed.

Fusc numbers are also known as: ::* fusc function (by Dijkstra, 1982) ::* Stern's Diatomic series (although it starts with unity, not zero) ::* Stern-Brocot sequence (although it starts with unity, not zero)

;Task: ::* show the first '''61''' fusc numbers (starting at zero) in a horizontal format. ::* show the fusc number (and its index) whose length is greater than any previous fusc number length. ::::* (the length is the number of digits when the fusc number is expressed in decimal.) ::* show all numbers with commas (if appropriate). ::* show all output here.

;Related task: ::* [[Stern-Brocot_sequence|RosettaCode Stern-Brocot sequence]]

;Also see: ::* the MathWorld entry: [http://mathworld.wolfram.com/SternsDiatomicSeries.html Stern's Diatomic Series]. ::* the OEIS entry: [http://oeis.org/A2487 A2487].

## ALGOL 68

```BEGIN
# calculate some members of the fusc sequence              #
#    f0 = 0, f1 = 1, fn = f(n/2)                 if n even #
#                       = f(n-1)/2) + f((n+1)/2) if n odd  #

# constructs an array of the first n elements of the fusc sequence #
PROC fusc sequence = ( INT n )[]INT:
BEGIN
[ 0 : n ]INT a;
IF n > 0 THEN
a[ 0 ] := 0;
IF n > 1 THEN
a[ 1 ] := 1;
INT i2 := 1;
FOR i FROM 2 BY 2 TO n - 1 DO
a[ i     ] := a[ i2 ];
a[ i + 1 ] := a[ # j - i # i2 ] + a[ # ( j + 1 ) OVER 2 # i2 + 1 ];
i2 +:= 1
OD
FI
FI;
a[ 0 : n - 1 AT 0 ]
END ; # fusc #

[]INT f = fusc sequence( 800 000 );
FOR i FROM 0 TO 60 DO print( ( " ", whole( f[ i ], 0 ) ) ) OD;
print( ( newline ) );
# find the lowest elements of the sequence that have 1, 2, 3, etc. digits #
print( ( "Sequence elements where number of digits of the value increase:", newline ) );
print( ( "       n    fusc(n)", newline ) );
INT digit power := 0;
FOR i FROM LWB f TO UPB f DO
IF f[ i ] >= digit power THEN
# found the first number with this many digits #
print( ( whole( i, -8 ), " ", whole( f[ i ], -10 ), newline ) );
IF digit power = 0 THEN digit power := 1 FI;
digit power *:= 10
FI
OD
END
```

{{out}}

```
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4
Sequence elements where number of digits of the value increase:
n    fusc(n)
0          0
37         11
1173        108
35499       1076
699051      10946

```

## AWK

```
# syntax: GAWK -f FUSC_SEQUENCE.AWK
# converted from C
BEGIN {
for (i=0; i<61; i++) {
printf("%d ",fusc(i))
}
printf("\n")
print("fusc numbers whose length is greater than any previous fusc number length")
printf("%9s %9s\n","fusc","index")
for (i=0; i<=700000; i++) {
f = fusc(i)
leng = num_leng(f)
if (leng > max_leng) {
max_leng = leng
printf("%9s %9s\n",commatize(f),commatize(i))
}
}
exit(0)
}
function commatize(x,  num) {
if (x < 0) {
return "-" commatize(-x)
}
x = int(x)
num = sprintf("%d.",x)
while (num ~ /^[0-9][0-9][0-9][0-9]/) {
sub(/[0-9][0-9][0-9][,.]/,",&",num)
}
sub(/\.\$/,"",num)
return(num)
}
function fusc(n) {
if (n == 0 || n == 1) {
return(n)
}
else if (n % 2 == 0) {
return fusc(n/2)
}
else {
return fusc((n-1)/2) + fusc((n+1)/2)
}
}
function num_leng(n,  sum) {
sum = 1
while (n > 9) {
n = int(n/10)
sum++
}
return(sum)
}

```

{{out}}

```
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4
fusc numbers whose length is greater than any previous fusc number length
fusc     index
0         0
11        37
108     1,173
1,076    35,499
10,946   699,051

```

## C

```
#include<limits.h>
#include<stdio.h>

int fusc(int n){
if(n==0||n==1)
return n;
else if(n%2==0)
return fusc(n/2);
else
return fusc((n-1)/2) + fusc((n+1)/2);
}

int numLen(int n){
int sum = 1;

while(n>9){
n = n/10;
sum++;
}

return sum;
}

void printLargeFuscs(int limit){
int i,f,len,maxLen = 1;

printf("\n\nPrinting all largest Fusc numbers upto %d \nIndex-------Value",limit);

for(i=0;i<=limit;i++){
f = fusc(i);
len = numLen(f);

if(len>maxLen){
maxLen = len;
printf("\n%5d%12d",i,f);
}
}
}

int main()
{
int i;

printf("Index-------Value");
for(i=0;i<61;i++)
printf("\n%5d%12d",i,fusc(i));
printLargeFuscs(INT_MAX);
return 0;
}

```

Prints first 61 Fusc numbers followed by the largest numbers :

```
Index-------Value
0           0
1           1
2           1
3           2
4           1
5           3
6           2
7           3
8           1
9           4
10           3
11           5
12           2
13           5
14           3
15           4
16           1
17           5
18           4
19           7
20           3
21           8
22           5
23           7
24           2
25           7
26           5
27           8
28           3
29           7
30           4
31           5
32           1
33           6
34           5
35           9
36           4
37          11
38           7
39          10
40           3
41          11
42           8
43          13
44           5
45          12
46           7
47           9
48           2
49           9
50           7
51          12
52           5
53          13
54           8
55          11
56           3
57          10
58           7
59          11
60           4

Printing all largest Fusc numbers upto 2147483647
Index-------Value
37          11
1173         108
35499        1076
699051      10946
103682   19573419
1010747  615164587

```

## C#

```using System;
using System.Collections.Generic;

static class program
{
static int n = 61;
static List<int> l = new List<int>() { 0, 1 };

static int fusc(int n)
{
if (n < l.Count) return l[n];
int f = (n & 1) == 0 ? l[n >> 1] : l[(n - 1) >> 1] + l[(n + 1) >> 1];
}

static void Main(string[] args)
{
bool lst = true; int w = -1, c = 0, t;
string fs = "{0,11:n0}  {1,-9:n0}", res = "";
Console.WriteLine("First {0} numbers in the fusc sequence:", n);
for (int i = 0; i < int.MaxValue; i++)
{
int f = fusc(i); if (lst)
{
if (i < 61) Console.Write("{0} ", f);
else
{
lst = false;
Console.WriteLine();
Console.WriteLine("Points in the sequence where an item has more digits than any previous items:");
Console.WriteLine(fs, "Index\\", "/Value"); Console.WriteLine(res); res = "";
}
}
if ((t = f.ToString().Length) > w)
{
w = t; res += (res == "" ? "" : "\n") + string.Format(fs, i, f);
if (!lst) { Console.WriteLine(res); res = ""; } if (++c > 5) break;
}
}
l.Clear();
}
}
```

{{out}}

```First 61 numbers in the fusc sequence:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4
Points in the sequence where an item has more digits than any previous items:
Index\  /Value
0  0
37  11
1,173  108
35,499  1,076
699,051  10,946
19,573,419  103,682
```

### The Function

```
// Generate the fusc sequence. Nigel Galloway: March 20th., 2019
let fG n=seq{for (n,g) in Seq.append n [1] |> Seq.pairwise do yield n; yield n+g}
let fusc=seq{yield 0; yield! Seq.unfold(fun n->Some(n,fG n))(seq[1])|>Seq.concat}|> Seq.mapi(fun n g->(n,g))

```

;Print first 62 elements

```
fusc |> Seq.take 61 |> Seq.iter(fun(_,g)->printf "%d " g); printfn ""

```

{{out}}

```
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4

```

;Show the fusc number (and its index) whose length is greater than any previous fusc number length The first 6 take only 10 secs so let me be more ambitious

```
let fN=let mutable n=0 in (fun (_,g)->if g>=n then n<-pown 10 (string g).Length; true else false)
fusc |> Seq.filter fN |> Seq.take 7 |> Seq.iter(fun(n,g)->printfn "fusc %d -> %d" n g)

```

{{out}}

```
fusc 0 -> 0
fusc 37 -> 11
fusc 1173 -> 108
fusc 35499 -> 1076
fusc 699051 -> 10946
fusc 19573419 -> 103682
fusc 615164587 -> 1010747
Real: 00:06:03.801, CPU: 00:06:03.140, GC gen0: 21336, gen1: 0

```

## Factor

```USING: arrays assocs formatting io kernel make math math.parser
math.ranges namespaces prettyprint sequences
tools.memory.private ;
IN: rosetta-code.fusc

<PRIVATE

: (fusc) ( n -- seq )
[ 2 ] dip [a,b) [
0 , 1 , [
[ building get ] dip dup even?
[ 2/ swap nth ]
[ [ 1 - 2/ ] [ 1 + 2/ ] 2bi [ swap nth ] 2bi@ + ]
if ,
] each
] { } make ;

: increases ( seq -- assoc )
[ 0 ] dip [
[
2array 2dup first number>string length <
[ [ 1 + ] [ , ] bi* ] [ drop ] if
] each-index
] { } make nip ;

PRIVATE>

: fusc ( n -- seq )
dup 3 < [ { 0 1 } swap head ] [ (fusc) ] if ;

: fusc-demo ( -- )
"First 61 fusc numbers:" print 61 fusc [ pprint bl ] each
nl nl
"Fusc numbers with more digits than all previous ones:"
print "Value   Index\n
### ===  ====
" print
1,000,000 fusc increases
[ [ commas ] bi@ "%-6s  %-7s\n" printf ] assoc-each ;

MAIN: fusc-demo
```

{{out}}

```
First 61 fusc numbers:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4

Fusc numbers with more digits than all previous ones:
Value   Index

### ===  ====

0       0
11      37
108     1,173
1,076   35,499
10,946  699,051

```

In [http://wiki.formulae.org/Fusc_sequence this] page you can see the solution of this task.

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text ([http://wiki.formulae.org/Editing_F%C5%8Drmul%C3%A6_expressions more info]). Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition.

The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.

## FreeBASIC

```' version 01-03-2019
' compile with: fbc -s console

#Define max 20000000

Dim Shared As UInteger f(max)

Sub fusc

f(0) = 0
f(1) = 1

For n As UInteger = 2 To max
If n And 1 Then
f(n) = f((n -1) \ 2) + f((n +1) \ 2)
Else
f(n) = f(n \ 2)
End If
Next

End Sub

' ------=< MAIN >=------

Dim As UInteger i, d
Dim As String fs

fusc

For i = 0 To 60
Print f(i); " ";
Next

Print : Print
Print "       Index       Value"
For i = 0 To max
If f(i) >= d Then
Print Using "###########," ; i; f(i)
If d = 0 Then d = 1
d *= 10
End If
Next

' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
```

{{out}}

```0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4

Index       Value
0           0
37          11
1,173         108
35,499       1,076
699,051      10,946
19,573,419     103,682
```

## Go

```package main

import (
"fmt"
"strconv"
)

func fusc(n int) []int {
if n <= 0 {
return []int{}
}
if n == 1 {
return []int{0}
}
res := make([]int, n)
res[0] = 0
res[1] = 1
for i := 2; i < n; i++ {
if i%2 == 0 {
res[i] = res[i/2]
} else {
res[i] = res[(i-1)/2] + res[(i+1)/2]
}
}
return res
}

func fuscMaxLen(n int) [][2]int {
maxLen := -1
maxFusc := -1
f := fusc(n)
var res [][2]int
for i := 0; i < n; i++ {
if f[i] <= maxFusc {
continue // avoid expensive strconv operation where possible
}
maxFusc = f[i]
le := len(strconv.Itoa(f[i]))
if le > maxLen {
res = append(res, [2]int{i, f[i]})
maxLen = le
}
}
return res
}

func commatize(n int) string {
s := fmt.Sprintf("%d", n)
if n < 0 {
s = s[1:]
}
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
if n >= 0 {
return s
}
return "-" + s
}

func main() {
fmt.Println("The first 61 fusc numbers are:")
fmt.Println(fusc(61))
fmt.Println("\nThe fusc numbers whose length > any previous fusc number length are:")
res := fuscMaxLen(20000000)  // examine first twenty million numbers say
for i := 0; i < len(res); i++ {
fmt.Printf("%7s (index %10s)\n", commatize(res[i][1]), commatize(res[i][0]))
}
}
```

{{out}}

```
The first 61 fusc numbers are:
[0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4]

The fusc numbers whose length > any previous fusc number length are:
0 (index          0)
11 (index         37)
108 (index      1,173)
1,076 (index     35,499)
10,946 (index    699,051)
103,682 (index 19,573,419)

```

```fusc :: Int -> Int
fusc i
| 1 > i = 0
| otherwise = fst \$ go (pred i)
where
go n
| 0 == n = (1, 0)
| even n = (x + y, y)
| otherwise = (x, x + y)
where
(x, y) = go (div n 2)

widths :: [(Int, Int)]
widths = (\(_, i, x) -> (i, x)) <\$> iterate nxtWidth (2, 0, 0)

nxtWidth :: (Int, Int, Int) -> (Int, Int, Int)
nxtWidth (w, i, v) =
let fi = (,) <*> fusc
(j, x) = until ((w <=) . length . show . snd) (fi . succ . fst) (fi i)
in (succ w, j, x)

main :: IO ()
main = do
putStrLn "First 61 terms:"
print \$ fusc <\$> [0 .. 60]
putStrLn "\n(Index, Value):"
mapM_ print \$ take 5 widths
```

{{Out}}

```First 61 terms:
[0,1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5,1,6,5,9,4,11,7,10,3,11,8,13,5,12,7,9,2,9,7,12,5,13,8,11,3,10,7,11,4]

(Index, Value):
(0,0)
(37,11)
(1173,108)
(35499,1076)
(699051,10946)
```

## J

```
fusc_term =: ({~ -:@#)`([: +/ ({~ ([: -: _1 1 + #)))@.(2 | #)
fusc =: (, fusc_term)@:]^:[ 0 1"_

NB. show the first 61 fusc numbers (starting at zero) in a horizontal format.
61 {. fusc 70
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4

9!:17]2 2 NB. specify bottom right position in box

FUSC =: fusc 99999
DIGITS =: ; ([: # 10&#.inv)&.> FUSC

(;: 'index value') ,. <"0(,: {&A) DIGITS i. 1 2 3 4
┌─────┬─┬──┬────┬─────┐
│index│0│37│1173│35499│
├─────┼─┼──┼────┼─────┤
│value│0│11│ 108│ 1076│
└─────┴─┴──┴────┴─────┘

```

## Javascript

### Functional

{{Trans|Python}}

A composition of pure generic functions:

```(() => {
'use strict';

const main = () => {

// fusc :: Int -> Int
const fusc = i => {
const go = n =>
0 === n ? (
[1, 0]
) : (() => {
const [x, y] = go(quot(n, 2));
return even(n) ? (
[x + y, y]
) : [x, x + y];
})();
return 1 > i ? (
0
) : fst(go(i - 1));
};

// firstWidths :: Int -> [(Int, Int)]
const firstWidths = n => {
const nxtWidth = xs => {
const
fi = fanArrow(fusc, id),
[x, j] = Array.from(until(
v => w <= fst(v).toString().length,
v => fi(succ(snd(v))),
fi(i)
));
return cons(
[succ(w), j, x],
xs
);
};
return until(
x => n < fst(fst(x)),
nxtWidth,
[[2, 0, 0]]
);
};

return unlines([
'First 61 terms:',
'[' + map(fusc, enumFromTo(0, 60)).join(',') + ']',
'',
'(Index, Value):',
unlines(map(
([i, x]) => '(' + i + ', ' + x + ')',
foldl(
(a, x) => cons(tail(x), a),
[],
firstWidths(5)
)
))
]);
};

// GENERIC FUNCTIONS ----------------------------

// Tuple (,) :: a -> b -> (a, b)
const Tuple = (a, b) => ({
type: 'Tuple',
'0': a,
'1': b,
length: 2
});

// cons :: a -> [a] -> [a]
const cons = (x, xs) =>
Array.isArray(xs) ? (
[x].concat(xs)
) : 'GeneratorFunction' !== xs.constructor.constructor.name ? (
x + xs
) : ( // Existing generator wrapped with one additional element
function*() {
yield x;
let nxt = xs.next()
while (!nxt.done) {
yield nxt.value;
nxt = xs.next();
}
}
)();

// enumFromTo :: Enum a => a -> a -> [a]
const enumFromTo = (m, n) => {
const [x, y] = [m, n].map(fromEnum),
b = x + ('number' !== typeof m ? 0 : m - x);
return Array.from({
length: 1 + (y - x)
}, (_, i) => toEnum(m)(b + i));
};

// even :: Int -> Bool
const even = n => 0 === n % 2;

// Compose a function from a simple value to a tuple of
// the separate outputs of two different functions

// fanArrow (&&&) :: (a -> b) -> (a -> c) -> (a -> (b, c))
const fanArrow = (f, g) => x => Tuple(f(x), g(x));

// foldl :: (a -> b -> a) -> a -> [b] -> a
const foldl = (f, a, xs) => xs.reduce(f, a);

// fromEnum :: Enum a => a -> Int
typeof x !== 'string' ? (
x.constructor === Object ? (
x.value
) : parseInt(Number(x))
) : x.codePointAt(0);

// fst :: (a, b) -> a
const fst = tpl => tpl[0];

// head :: [a] -> a
const head = xs => xs.length ? xs[0] : undefined;

// id :: a -> a
const id = x => x;

// map :: (a -> b) -> [a] -> [b]
const map = (f, xs) =>
(Array.isArray(xs) ? (
xs
) : xs.split('')).map(f);

// quot :: Int -> Int -> Int
const quot = (n, m) => Math.floor(n / m);

// snd :: (a, b) -> b
const snd = tpl => tpl[1];

// succ :: Enum a => a -> a
const succ = x => {
const t = typeof x;
return 'number' !== t ? (() => {
const [i, mx] = [x, maxBound(x)].map(fromEnum);
return i < mx ? (
toEnum(x)(1 + i)
) : Error('succ :: enum out of range.')
})() : x < Number.MAX_SAFE_INTEGER ? (
1 + x
) : Error('succ :: Num out of range.')
};

// tail :: [a] -> [a]
const tail = xs => 0 < xs.length ? xs.slice(1) : [];

// The first argument is a sample of the type
// allowing the function to make the right mapping

// toEnum :: a -> Int -> a
const toEnum = e => x => {
const
m = e.enum,
f = {
'number': Number,
'string': String.fromCodePoint,
'boolean': Boolean
} [typeof e];
return f ? (
f(x)
) : m[m[x]];
};

// unlines :: [String] -> String
const unlines = xs => xs.join('\n');

// until :: (a -> Bool) -> (a -> a) -> a -> a
const until = (p, f, x) => {
let v = x;
while (!p(v)) v = f(v);
return v;
};

// MAIN ---
return main();
})();
```

{{Out}}

```First 61 terms:
[0,1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,1,5,4,7,3,8,5,7,2,7,5,8,3,7,4,5,1,6,5,9,4,11,7,10,3,11,8,13,5,12,7,9,2,9,7,12,5,13,8,11,3,10,7,11,4]

(Index, Value):
(0, 0)
(37, 11)
(1173, 108)
(35499, 1076)
(699051, 10946)
```

## Julia

```using Memoize, Formatting

@memoize function sternbrocot(n)
if n < 2
return n
elseif iseven(n)
return sternbrocot(div(n, 2))
else
m = div(n - 1, 2)
return sternbrocot(m) + sternbrocot(m + 1)
end
end

function fusclengths(N=100000000)
println("sequence number : fusc value")
maxlen = 0
for i in 0:N
x = sternbrocot(i)
if (len = length(string(x))) > maxlen
println(lpad(format(i, commas=true), 15), " : ", format(x, commas=true))
maxlen = len
end
end
end

println("The first 61 fusc numbers are: ", [sternbrocot(x) for x in 0:60])
fusclengths()

```

{{out}}

```
The first 61 fusc numbers are: [0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6,
5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4]
sequence number : fusc value
0 : 0
37 : 11
1,173 : 108
35,499 : 1,076
699,051 : 10,946
19,573,419 : 103,682

```

## Kotlin

{{trans|Go}}

```// Version 1.3.21

fun fusc(n: Int): IntArray {
if (n <= 0) return intArrayOf()
if (n == 1) return intArrayOf(0)
val res = IntArray(n)
res[1] = 1
for (i in 2 until n) {
if (i % 2 == 0) {
res[i] = res[i / 2]
} else {
res[i] = res[(i - 1) / 2] + res[(i + 1) / 2]
}
}
return res
}

fun fuscMaxLen(n: Int): List<Pair<Int, Int>> {
var maxLen = -1
var maxFusc = -1
val f = fusc(n)
val res = mutableListOf<Pair<Int, Int>>()
for (i in 0 until n) {
if (f[i] <= maxFusc) continue // avoid string conversion
maxFusc = f[i]
val len = f[i].toString().length
if (len > maxLen) {
maxLen = len
}
}
return res
}

fun main() {
println("The first 61 fusc numbers are:")
println(fusc(61).asList())
println("\nThe fusc numbers whose length > any previous fusc number length are:")
val res = fuscMaxLen(20_000_000)  // examine first 20 million numbers say
for (r in res) {
System.out.printf("%,7d (index %,10d)\n", r.second, r.first)
}
}
```

{{output}}

```
The first 61 fusc numbers are:
[0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4]

The fusc numbers whose length > any previous fusc number length are:
0 (index          0)
11 (index         37)
108 (index      1,173)
1,076 (index     35,499)
10,946 (index    699,051)
103,682 (index 19,573,419)

```

## Pascal

{{works with|Free Pascal}} Using dynamic array.To speed things up using Pointer. Found the indices of a specific base to oszillating.Tried power of phi with more success 11 ~ phi^5

```program fusc;
uses
sysutils;
const
MaxIdx =1253*1000*1000;//19573420; // must be even
type
tFuscElem = LongWord;
tFusc = array of tFuscElem;
var
FuscField : tFusc;

function commatize(n:NativeUint):string;
var
l,i : NativeUint;
begin
str(n,result);
l := length(result);
//no commatize
if l < 4 then
exit;
//new length
i := l+ (l-1) DIV 3;
setlength(result,i);
//copy chars to the right place
While i <> l do
Begin
result[i]:= result[l];result[i-1]:= result[l-1];
result[i-2]:= result[l-2];result[i-3]:= ',';
dec(i,4);dec(l,3);
end;
end;

procedure OutFusc(StartIdx,EndIdx :NativeInt;const FF:tFusc);
Begin
IF StartIdx < Low(FF) then StartIdx :=Low(FF);
IF EndIdx > High(FF) then EndIdx := High(FF);
For StartIdx := StartIdx to EndIdx do
write(FF[StartIdx],' ');
writeln;
end;

procedure FuscCalc(var FF:tFusc);
var
pFFn,pFFi : ^tFuscElem;
i,n,sum : NativeUint;
Begin
FF[0]:= 0;
FF[1]:= 1;
n := 2;
i := 1;
pFFn := @FF[n];
pFFi := @FF[i];
sum := pFFi^;
while n <= MaxIdx-2 do
begin
//even
pFFn^ := sum;//FF[n] := FF[i];
//odd
inc(pFFi);//FF[i+1]
inc(pFFn);//FF[n+1]
sum := sum+pFFi^;
pFFn^:= sum; //FF[n+1] := FF[i]+FF[i+1];
sum := pFFi^;
inc(pFFn);
inc(n,2);
//inc(i);
end;
end;

begin
writeln('Fusc numbers with more digits in base ',base,' than all previous ones:');
writeln('Value':10,'Index':10,'  IndexNum/IndexNumBefore');
writeln('======':10,'
### =
':14);
end;

procedure CheckFuscDigits(const FF:tFusc;Base:NativeUint);
var
pFF : ^tFuscElem;
Dig,
i,lastIdx: NativeInt;
Begin
Dig := -1;
i := 0;
lastIdx := 0;
pFF := @FF[0];// aka FF[i]
repeat
//search in tight loop speeds up
repeat
inc(pFF);
inc(i);
until pFF^ >Dig;

if i>= MaxIdx then
BREAK;
//output
write(commatize(pFF^):10,commatize(i):14);//,DIG:10);
IF lastIdx> 0 then
write(i/lastIdx:12:7);
writeln;
lastIdx := i;
IF Dig >0 then
Dig := Dig*Base+Base-1
else
Dig := Base-1;
until false;
writeln;
end;

BEGIN
setlength(FuscField,MaxIdx);
FuscCalc(FuscField);
writeln('First 61 fusc numbers:');
OutFusc(0,60,FuscField);

CheckFuscDigits(FuscField,10);
CheckFuscDigits(FuscField,11); //11 ~phi^5  1.6180..^5 = 11,09
setlength(FuscField,0);
END.
```

{{Out}}

```First 61 fusc numbers:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4
Fusc numbers with more digits in base 10 than all previous ones:
Value     Index  IndexNum/IndexNumBefore

### ===       ====

1             1
11            37  37.0000000
108         1,173  31.7027027
1,076        35,499  30.2634271
10,946       699,051  19.6921322
103,682    19,573,419  27.9999871
1,010,747   615,164,587  31.4285709

Fusc numbers with more digits in base 11 than all previous ones:
Value     Index  IndexNum/IndexNumBefore

### ===       ====

1             1
11            37  37.0000000
123         1,195  32.2972973
1,364        38,229  31.9907950
15,127     1,223,339  32.0002877
167,761    39,146,837  31.9999910
1,860,498 1,252,698,795  32.0000003

real  0m1,968s  user  0m1,594s  sys 0m0,373s
```

## Perl

Borrowing from the [http://rosettacode.org/wiki/Stern-Brocot_sequence Stern-Brocot sequence] task.

```use strict;
use warnings;
use feature 'say';

sub comma { reverse ((reverse shift) =~ s/(.{3})/\$1,/gr) =~ s/^,//r }

sub stern_diatomic {
my (\$p,\$q,\$i) = (0,1,shift);
while (\$i) {
if (\$i & 1) { \$p += \$q; } else { \$q += \$p; }
\$i >>= 1;
}
\$p;
}

say "First 61 terms of the Stern-Brocot sequence:\n" . join ' ', map { stern_diatomic(\$_) } 0..60;
say "\nIndex and value for first term longer than any previous:";

my \$i =  0;
my \$l = -1;
while (\$l < 5) {
my \$v = stern_diatomic(\$i);
printf("%15s : %s\n", comma(\$i), comma(\$v)) and \$l = length \$v if length \$v > \$l;
\$i++;
}
```

{{out}}

```First 61 terms of the Stern-Brocot sequence:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4

Index and value for first term longer than any previous:
0 : 0
37 : 11
1,173 : 108
35,499 : 1,076
699,051 : 10,946
```

## Perl 6

{{works with|Rakudo|2018.12}}

```my @Stern-Brocot;
@Stern-Brocot = 0, 1, 1, { |(@Stern-Brocot[\$_ - 1] + @Stern-Brocot[\$_], @Stern-Brocot[\$_]) given ++\$+1 } ... *;

sub comma { \$^i.flip.comb(3).join(',').flip }

put "First 61 terms of the Stern-Brocot sequence:\n{@Stern-Brocot[^61].gist}" ~
"\n\nIndex and value for first term longer than any previous:";

for flat 'Index', 'Value', 0, 0, (1..4).map({
my \$l = 10**\$_;
@Stern-Brocot.first(* > \$l, :kv).map: *.&comma
}) -> \$i, \$v {
printf "%15s : %s\n", \$i, \$v
}
```

{{out}}

```First 61 terms of the Stern-Brocot sequence:
(0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4)

Index and value for first term longer than any previous:
Index : Value
0 : 0
37 : 11
1,173 : 108
35,499 : 1,076
699,051 : 10,946

```

## Phix

Note that phix is 1-indexed. While there are no commas in the first 61 entries, it felt more in line with the task requirements to forego the standard comma-separated %v output.

```constant limit = 20_000_000
sequence fuscs = repeat(0,limit); -- NB 1-based indexing; fusc(0)===fuscs[1]
fuscs[2] = 1                                        -- ie fusc(1):=1
for n=3 to limit do
fuscs[n] = iff(remainder(n-1,2)?fuscs[n/2]+fuscs[n/2+1]:fuscs[(n+1)/2])
end for
--printf(1,"First 61 terms of the Fusc sequence:\n%v\n",{fuscs[1..61]})
string s = ""
for n=1 to 61 do s&=sprintf("%,d ",fuscs[n]) end for
printf(1,"First 61 terms of the Fusc sequence:\n%s\n\n",{s})
printf(1,"Elements with more digits than any previous items:\n")
printf(1,"          Index : Value\n")
integer d = 0
for n=1 to length(fuscs) do
if fuscs[n]>=d then
printf(1,"%,15d : %,d\n",{n-1,fuscs[n]})
d = iff(d=0?10:d*10)
end if
end for
```

{{out}}

```
First 61 terms of the Fusc sequence:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4

Elements with more digits than any previous items:
Index : Value
0 : 0
37 : 11
1,173 : 108
35,499 : 1,076
699,051 : 10,946
19,573,419 : 103,682

```

## Python

By composition of pure functions, for better reliability, ease and speed of refactoring, and for higher levels of code reuse,

```"""Fusc sequence and tests"""

# fusc :: Int -> Int
def fusc(i):
'''Fusc sequence'''
def go(n):
if 0 == n:
return (1, 0)
else:
x, y = go(n // 2)
return (x + y, y) if 0 == n % 2 else (
x, x + y
)
return 0 if 1 > i else (
go(i - 1)[0]
)

# main :: IO()
def main():
'''Tests'''

print('First 61 terms:')
print(
list(map(fusc, range(0, 61)))
)
print('\n(Index, Value):')

# Up to five digits
for tpl in firstWidths(5):
print(tpl)

# firstWidths :: Int -> [(Int, Int)]
def firstWidths(n):
'''First terms to have particular widths (digit counts) up to n'''

# nxtFusc :: (Int, Int) -> (Int, Int)
def nxtFusc(tpl):
i = 1 + tpl[1]
return (fusc(i), i)

# nxtWidth :: [(Int, Int, Int)] -> [(Int, Int, Int)]
def nxtWidth(xs):
'''(width, index, value)'''
w, i, _ = xs[0]

def p(tpl):
return w <= len(str(tpl[0]))
x, j = until(p)(nxtFusc)(
(fusc(i), i)
)
return [(1 + w, j, x)] + xs

# go :: Int -> [(Int, Int, Int)]
def go(n):
def p(xs):
return n <= len(xs)
return until(p)(nxtWidth)(
[(2, 0, 0)]
)
return list(map(tail, reversed(go(n))))

# GENERIC -------------------------------------------------

# tail :: [a] -> [a]
def tail(xs):
'''The elements following the head of a (non-empty) list.'''
return xs[1:]

# until :: (a -> Bool) -> (a -> a) -> a -> a
def until(p):
'''The result of applying f until p holds.
The initial seed value is x.'''
def go(f, x):
v = x
while not p(v):
v = f(v)
return v
return lambda f: lambda x: go(f, x)

# TEST ----------------------------------------------------
if __name__ == '__main__':
main()
```

{{Out}}

```First 61 terms:
[0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4]

(Index, Value):
(0, 0)
(37, 11)
(1173, 108)
(35499, 1076)
(699051, 10946)
```

## Racket

```#lang racket

(require racket/generator)

(define (memoize f)
(define table (make-hash))
(λ args (hash-ref! table args (thunk (apply f args)))))

(define fusc
(memoize
(λ (n)
(cond
[(<= n 1) n]
[(even? n) (fusc (/ n 2))]
[else (+ (fusc (/ (sub1 n) 2)) (fusc (/ (add1 n) 2)))]))))

(define (comma x)
(string-join
(reverse
(for/list ([digit (in-list (reverse (string->list (~a x))))] [i (in-naturals)])
(cond
[(and (= 0 (modulo i 3)) (> i 0)) (string digit #\,)]
[else (string digit)])))
""))

(displayln (string-join (for/list ([i (in-range 61)]) (comma (fusc i))) " "))
(newline)

(define gen
(in-generator
(let loop ([prev 0] [i 0])
(define result (fusc i))
(define len (string-length (~a result)))
(cond
[(> len prev)
(yield (list i result))

(for ([i (in-range 5)] [x gen])
(match-define (list index result) x)
(printf "~a: ~a\n" (comma index) (comma result)))
```

{{out}}

```
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4

0: 0
37: 11
1,173: 108
35,499: 1,076
699,051: 10,946

```

## REXX

```/*REXX program  calculates and displays the   fusc   (or  Stern's Diatomic)   sequence. */
parse arg LO HI xw .                             /*obtain optional arguments from the CL*/
if LO=='' | LO==","  then LO=  0                 /*Not specified?  Then use the default.*/
if HI=='' | HI==","  then HI= 61                 /* "      "         "   "   "     "    */
if xw=='' | xw==","  then xw=  0                 /* "      "         "   "   "     "    */
list= xw<1                                       /*boolean value:  LIST  to show numbers*/
@.=;        @.0= 0;       @.1= 1                 /*assign array default; assign low vals*/
mL= 0                                            /*the maximum length (digits)  so far. */
\$=                                               /* "  list of  fusc  numbers    "  "   */
do j=0  for HI+1                              /*process a bunch of integers from zero*/
if j>1  then if j//2  then do;  _= (j-1) % 2;   p= (j+1) % 2;   @.j= @._ + @.p;   end
else do;  _= j % 2;                       @.j= @._;         end
if list  then if j>=LO  then \$= \$ commas(@.j)                      /*add it to a list*/
else nop
else do;   if length(@.j)<=mL  then iterate               /*still too small.*/
mL= length(@.j)                                /*found increase. */
if mL==1  then say '═══index═══   ═══fusc number═══'
say right( commas(j), 9)     right( commas(@.j), 14)
if mL==xw  then leave     /*Found max length?  Then stop looking.*/
end                             /* [↑]  display fusc #s of maximum len.*/
end   /*j*/

if \$\==''  then say strip(\$)                     /*display a horizontal list of fusc #s.*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas:  parse arg _;  do c=length(_)-3  to 1  by -3; _=insert(',', _, c); end;   return _
```

{{out|output|text= when using the default inputs:}}

```
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4 9

```

{{out|output|text= when using the inputs of: 0 999999999 5 }}

```
═══index═══   ═══fusc number═══
0              0
37             11
1,173            108
35,499          1,076
699,051         10,946

```

## Ring

```
# Project: Fusc sequence

max = 60
fusc = list(36000)
fusc[1] = 1
see "working..." + nl
see "wait for done..." + nl
see "The first 61 fusc numbers are:" + nl
fuscseq(max)
see "0"
for m = 1 to max
see " " + fusc[m]
next

see nl
see "The fusc numbers whose length > any previous fusc number length are:" + nl
see "Index Value" + nl
see " 0     0" + nl
d = 10
for i = 1 to 36000
if fusc[i] >= d
see " " + i + "   " + fusc[i] + nl
if d = 0
d = 1
ok
d = d*10
ok
next
see "done..." + nl

func fuscseq(max)
for n = 2 to 36000
if n%2 = 1
fusc[n] = fusc[(n-1)/2] + fusc[(n+1)/2]
but n%2 = 0
fusc[n] = fusc[n/2]
ok
next

```

{{out}}

```
working...
wait for done...
The first 61 fusc numbers are:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4
The fusc numbers whose length > any previous fusc number length are:
Index Value
0     0
37    11
1173  108
35499 1076
done...

```

## Ruby

Using two Enumerators; the second making use of the first:

```fusc = Enumerator.new do |y|
y << 0
y << 1
arr = [0,1]
2.step do |n|
res = n.even? ? arr[n/2] : arr[(n-1)/2] + arr[(n+1)/2]
y   << res
arr << res
end
end

fusc_max_digits = Enumerator.new do |y|
cur_max, cur_exp = 0, 0
0.step do |i|
f = fusc.next
if f >= cur_max
cur_exp += 1
cur_max = 10**cur_exp
y << [i, f]
end
end
end

puts fusc.take(61).join(" ")
fusc_max_digits.take(6).each{|pair| puts "%15s : %s" % pair }

```

{{out}}

```0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4
0 : 0
11 : 37
108 : 1173
1076 : 35499
10946 : 699051
103682 : 19573419

```

## Sidef

```func fusc(n) is cached {

return 0 if n.is_zero
return 1 if n.is_one

n.is_even ? fusc(n/2) : (fusc((n-1)/2) + fusc(((n-1)/2)+1))
}

say ("First 61 terms of the Stern-Brocot sequence: ", 61.of(fusc).join(' '))

say "\nIndex and value for first term longer than any previous:"
printf("%15s : %s\n", "Index", "Value");

var (index=0, len=0)

5.times {
index = (index..Inf -> first_by { fusc(_).len > len })
len = fusc(index).len
printf("%15s : %s\n", index.commify, fusc(index).commify)
}
```

{{out}}

```
First 61 terms of the Stern-Brocot sequence: 0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4

Index and value for first term longer than any previous:
Index : Value
0 : 0
37 : 11
1,173 : 108
35,499 : 1,076
699,051 : 10,946

```

## Visual Basic .NET

{{trans|C#}}

```Module Module1

Dim n As Integer = 61, l As List(Of Integer) = {0, 1}.ToList

Function fusc(n As Integer) As Integer
If n < l.Count Then Return l(n)
fusc = If((n And 1) = 0, l(n >> 1), l((n - 1) >> 1) + l((n + 1) >> 1))
End Function

Sub Main(args As String())
Dim lst As Boolean = True, w As Integer = -1, c As Integer = 0,
fs As String = "{0,11:n0}  {1,-9:n0}", res As String = ""
Console.WriteLine("First {0} numbers in the fusc sequence:", n)
For i As Integer = 0 To Integer.MaxValue
Dim f As Integer = fusc(i)
If lst Then
If i < 61 Then
Console.Write("{0} ", f)
Else
lst = False
Console.WriteLine()
Console.WriteLine("Points in the sequence where an item has more digits than any previous items:")
Console.WriteLine(fs, "Index\", "/Value") : Console.WriteLine(res) : res = ""
End If
End If
Dim t As Integer = f.ToString.Length
If t > w Then
w = t
res &= If(res = "", "", vbLf) & String.Format(fs, i, f)
If Not lst Then Console.WriteLine(res) : res = ""
c += 1 : If c > 5 Then Exit For
End If
Next : l.Clear()
End Sub
End Module

```

{{out}}

```First 61 numbers in the fusc sequence:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4
Points in the sequence where an item has more digits than any previous items:
Index\  /Value
0  0
37  11
1,173  108
35,499  1,076
699,051  10,946
19,573,419  103,682

```

## zkl

```fuscs:=List.createLong(1_000_000, 0); fuscs[1]=1; // we'll just use a big count
foreach n in ([2..fuscs.len()-1]){		 // and generate
fuscs[n]=( if(n.isEven()) fuscs[n/2] else fuscs[(n-1)/2] + fuscs[(n+1)/2] )
}

println("First 61 terms of the Stern-Brocot sequence:");
fuscs[0,61].concat(" ").println();

println("\nIndex and value for first term longer than any previous:");
println("          Index : Value");
prevMax:=-1;
foreach n in (fuscs.len()){
f,fd := fuscs[n], f.numDigits;
if(fd>prevMax){ println("%15,d : %,d".fmt(n,f)); prevMax=fd }
}
```

{{out}}

```
First 61 terms of the Stern-Brocot sequence:
0 1 1 2 1 3 2 3 1 4 3 5 2 5 3 4 1 5 4 7 3 8 5 7 2 7 5 8 3 7 4 5 1 6 5 9 4 11 7 10 3 11 8 13 5 12 7 9 2 9 7 12 5 13 8 11 3 10 7 11 4

Index and value for first term longer than any previous:
Index : Value
0 : 0
37 : 11
1,173 : 108
35,499 : 1,076
699,051 : 10,946

```