# Fusc sequence

Fusc sequence
You are encouraged to solve this task according to the task description, using any language you may know.

Definitions

The   fusc   integer sequence is defined as:

•   fusc(0) = 0
•   fusc(1) = 1
•   for n>1,   the   nth   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)

•   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.

Also see

## 11l

Translation of: Kotlin

<lang 11l>F fusc(n)

```  V res = [0] * n
res[1] = 1
L(i) 2 .< n
res[i] = I i % 2 == 0 {res[i I/ 2]} E res[(i-1) I/ 2] + res[(i+1) I/ 2]
R res
```

print(‘First 61 terms:’) print(fusc(61))

print() print(‘Points in the sequence where an item has more digits than any previous items:’) V f = fusc(20'000'000) V max_len = 0 L(i) 0 .< f.len

```  I String(f[i]).len > max_len
max_len = String(f[i]).len
print((i, f[i]))</lang>
```
Output:
```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]

Points in the sequence where an item has more digits than any previous items:
(0, 0)
(37, 11)
(1173, 108)
(35499, 1076)
(699051, 10946)
(19573419, 103682)
```

procedure Show_Fusc is

```  generic
Precalculate : Natural;
package Fusc_Sequences is
function Fusc (N : in Natural) return Natural;
end Fusc_Sequences;
```
```  package body Fusc_Sequences is
```
```     Precalculated_Fusc : array (0 .. Precalculate) of Natural;
```
```     function Fusc_Slow (N : in Natural) return Natural is
begin
if N = 0 or N = 1 then
return N;
elsif N mod 2 = 0 then
return Fusc_Slow (N / 2);
else
return Fusc_Slow ((N - 1) / 2) + Fusc_Slow ((N + 1) / 2);
end if;
end Fusc_Slow;
```
```     function Fusc (N : in Natural) return Natural is
begin
if N <= Precalculate then
return Precalculated_Fusc (N);
elsif N mod 2 = 0 then
return Fusc (N / 2);
else
return Fusc ((N - 1) / 2) + Fusc ((N + 1) / 2);
end if;
end Fusc;
```
```  begin
for N in Precalculated_Fusc'Range loop
Precalculated_Fusc (N) := Fusc_Slow (N);
end loop;
end Fusc_Sequences;
```

```  package Fusc_Sequence is
new Fusc_Sequences (Precalculate => 200_000);
```
```  function Fusc (N : in Natural) return Natural
renames Fusc_Sequence.Fusc;
```

```  procedure Print_Small_Fuscs is
begin
Put_Line ("First 61 numbers in the fusc sequence:");
for N in 0 .. 60 loop
Put (Fusc (N)'Image);
Put (" ");
end loop;
New_Line;
end Print_Small_Fuscs;
```

```  procedure Print_Large_Fuscs (High : in Natural) is
subtype N_Range is Natural range Natural'First .. High;
F       : Natural;
Len     : Natural;
Max_Len : Natural := 0;
Placeholder : String := "       n      fusc(n)";
Image_N     : String renames Placeholder (1  .. 8);
Image_Fusc  : String renames Placeholder (10 .. Placeholder'Last);
begin
New_Line;
Put_Line ("Printing all largest Fusc numbers upto " & High'Image);
Put_Line (Placeholder);
```
```     for N in N_Range loop
F   := Fusc (N);
Len := F'Image'Length;
```
```         if Len > Max_Len then
Max_Len := Len;
Put (Image_N,    N);
Put (Image_Fusc, F);
Put (Placeholder);
New_Line;
end if;
end loop;
end Print_Large_Fuscs;
```

begin

```  Print_Small_Fuscs;
Print_Large_Fuscs (High => 20_000_000);
```

end Show_Fusc;</lang>

Output:
```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

Printing all largest Fusc numbers upto  20000000
n      fusc(n)
0            0
37           11
1173          108
35499         1076
699051        10946
19573419       103682```

## ALGOL 68

<lang algol68>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</lang>

Output:
``` 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
```

## AppleScript

<lang applescript>on fusc(n)

```   if (n < 2) then
return n
else if (n mod 2 is 0) then
return fusc(n div 2)
else
return fusc((n - 1) div 2) + fusc((n + 1) div 2)
end if
```

end fusc

set sequence to {} set longestSoFar to 0 repeat with i from 0 to 60

```   set fuscNumber to fusc(i)
set end of sequence to fuscNumber
set len to (count (fuscNumber as text))
if (len > longestSoFar) then
set longestSoFar to len
set firstLongest to fuscNumber
set indexThereof to i + 1 -- AppleScript indices are 1-based.
end if
```

end repeat

return {sequence:sequence, firstLongest:firstLongest, indexThereof:indexThereof}</lang>

Output:
`{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}, firstLongest:11, indexThereof:38}`

## AWK

<lang AWK>

1. syntax: GAWK -f FUSC_SEQUENCE.AWK
2. 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)
```

} </lang>

Output:
```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

<lang C>

1. include<limits.h>
2. 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;
```

} </lang> 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#

<lang csharp>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();
}
```

}</lang>

Output:
```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 ```

## C++

Translation of: C#

<lang cpp>#include <iomanip>

1. include <iostream>
2. include <limits>
3. include <sstream>
4. include <vector>

const int n = 61; std::vector<int> l{ 0, 1 };

int fusc(int n) {

```   if (n < l.size()) return l[n];
int f = (n & 1) == 0 ? l[n >> 1] : l[(n - 1) >> 1] + l[(n + 1) >> 1];
l.push_back(f);
return f;
```

}

int main() {

```   bool lst = true;
int w = -1;
int c = 0;
int t;
std::string res;
std::cout << "First " << n << " numbers in the fusc sequence:\n";
for (int i = 0; i < INT32_MAX; i++) {
int f = fusc(i);
if (lst) {
if (i < 61) {
std::cout << f << ' ';
} else {
lst = false;
std::cout << "\nPoints in the sequence where an item has more digits than any previous items:\n";
std::cout << std::setw(11) << "Index\\" << "  " << std::left << std::setw(9) << "/Value\n";
std::cout << res << '\n';
res = "";
}
}
std::stringstream ss;
ss << f;
t = ss.str().length();
ss.str("");
ss.clear();
if (t > w) {
w = t;
res += (res == "" ? "" : "\n");
ss << std::setw(11) << i << "  " << std::left << std::setw(9) << f;
res += ss.str();
if (!lst) {
std::cout << res << '\n';
res = "";
}
if (++c > 5) {
break;
}
}
}
return 0;
```

}</lang>

Output:
```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
1173  108
35499  1076
699051  10946
19573419  103682```

## D

<lang d>import std.conv; import std.format; import std.stdio;

enum n = 61; int[] l = [0, 1];

int fusc(int n) {

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

}

void main() {

```   bool lst = true;
int w = -1;
int c = 0;
int t;
string fs = "%11s  %-9s";
string res = "";
for (int i = 0; i < int.max; i++) {
int f = fusc(i);
if (lst) {
if (i < 61) {
write(f, ' ');
} else {
lst = false;
writeln;
writeln("Points in the sequence where an item has more digits than any previous items:");
writefln(fs, "Index\\", "/Value");
writeln(res);
res = "";
}
}
t = f.to!string.length;
if (t > w) {
w = t;
res ~= (res == "" ? "" : "\n") ~ format(fs, i, f);
if (!lst) {
writeln(res);
res = "";
}
if (++c > 5) {
break;
}
}
}
l.length = 0;
```

}</lang>

Output:
```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
1173  108
35499  1076
699051  10946
19573419  103682```

## F#

### The Function

<lang fsharp> // 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)) </lang>

Print first 62 elements

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

Output:
```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 <lang fsharp> 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) </lang>

Output:
```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

<lang 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</lang>

Output:
```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
```

## FreeBASIC

<lang freebasic>' version 01-03-2019 ' compile with: fbc -s console

1. 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</lang>

Output:
```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```

## Fōrmulæ

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text (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.

## Go

<lang 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]))
}
```

}</lang>

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)
```

<lang haskell>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</lang>
```
Output:
```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

<lang 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│ └─────┴─┴──┴────┴─────┘

</lang>

## Java

<lang Java>

public class FuscSequence {

```   public static void main(String[] args) {
System.out.println("Show the first 61 fusc numbers (starting at zero) in a horizontal format");
for ( int n = 0 ; n < 61 ; n++ ) {
System.out.printf("%,d ", fusc[n]);
}

System.out.printf("%n%nShow the fusc number (and its index) whose length is greater than any previous fusc number length.%n");
int start = 0;
for (int i = 0 ; i <= 5 ; i++ ) {
int val = i != 0 ? (int) Math.pow(10, i) : -1;
for ( int j = start ; j < FUSC_MAX ; j++ ) {
if ( fusc[j] > val ) {
System.out.printf("fusc[%,d] = %,d%n", j, fusc[j] );
start = j;
break;
}
}
}
}

private static final int FUSC_MAX = 30000000;
private static int[] fusc = new int[FUSC_MAX];
```
```   static {
fusc[0] = 0;
fusc[1] = 1;
for ( int n = 2 ; n < FUSC_MAX ; n++ ) {
fusc[n] = (n % 2 == 0 ? fusc[n/2] : fusc[(n-1)/2] + fusc[(n+1)/2]);
}
}
```

} </lang>

Output:
```Show the first 61 fusc numbers (starting at zero) in a horizontal format
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.
fusc[0] = 0
fusc[37] = 11
fusc[1,173] = 108
fusc[35,499] = 1,076
fusc[699,051] = 10,946
fusc[19,573,419] = 103,682
```

## JavaScript

### Functional

Translation of: Python

A composition of pure generic functions: <lang javascript>(() => {

```   '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();
```

})();</lang>

Output:
```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

<lang 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()

</lang>
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]
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

Translation of: Go

<lang scala>// 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)
}
```

}</lang>

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)
```

## Nim

<lang nim>import strformat

func fusc(n: int): int =

``` if n == 0 or n == 1:
n
elif n mod 2 == 0:
fusc(n div 2)
else:
fusc((n - 1) div 2) + fusc((n + 1) div 2)
```

echo "The first 61 fusc numbers:" for i in 0..61:

``` write(stdout, fmt"{fusc(i)} ")
```

echo "\n\nThe fusc numbers whose lengths are greater than those of previous fusc numbers:" echo fmt" n fusc(n)" echo "--------- ---------" var maxLength = 0 for i in 0..700_000:

``` var f = fusc(i)
var length = len(\$f)
if length > maxLength:
maxLength = length
echo fmt"{i:9} {f:9}"</lang>
```
Output:
```The 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 9

The fusc numbers whose lengths are greater than those of previous fusc numbers:
n   fusc(n)
--------- ---------
0         0
37        11
1173       108
35499      1076
699051     10946
```

## 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 <lang pascal>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;

``` 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

``` OutHeader(base);
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.</lang>

Output:
```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 Stern-Brocot sequence task. <lang perl>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++;
```

}</lang>

Output:
```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```

## 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. <lang Phix>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</lang>

Output:
```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

### Procedural

<lang python>from collections import deque from itertools import islice, count

def fusc():

```   q = deque([1])
yield 0
yield 1
```
```   while True:
x = q.popleft()
q.append(x)
yield x
```
```       x += q[0]
q.append(x)
yield x
```

def longest_fusc():

```   sofar = 0
for i, f in zip(count(), fusc()):
if f >= sofar:
yield(i, f)
sofar = 10 * sofar or 10
```

print('First 61:') print(list(islice(fusc(), 61)))

print('\nLength records:') for i, f in islice(longest_fusc(), 6):

```   print(f'fusc({i}) = {f}')
```

</lang>

Output:
```First 61:
[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]

Length records:
fusc(0) = 0
fusc(37) = 11
fusc(1173) = 108
fusc(35499) = 1076
fusc(699051) = 10946
fusc(19573419) = 103682```

### Functional

By composition of pure functions, avoiding mutable variables, and confining any unavoidables to the internals of well-tested primitives: <lang python>Fusc sequence

from itertools import chain, count, islice from operator import itemgetter

1. As an infinite stream of terms,
1. infiniteFusc :: [Int]

def infiniteFusc():

```   Fusc sequence.
OEIS A2487

def go(step):
isEven, n, xxs = step
x, xs = xxs[0], xxs[1:]
if isEven:
nxt = n + x
return not isEven, nxt, xxs + [nxt]
else:
return not isEven, x, xs + [x]
```
```   return chain(
[0, 1],
map(
itemgetter(1),
iterate(go)(
(True, 1, [1])
)
)
)
```

1. Or as a function over an integer:
1. 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]
)
```

1. firstFuscOfEachMagnitude ::

def firstFuscOfEachMagnitude():

```   Non-finite stream of each term
in OEIS A2487 that requires an
unprecedented quantity of decimal digits.

a2487 = enumerate(map(fusc, count()))
```
```   def go(e):
limit = 10 ** e
return next(
(i, x) for i, x in a2487
if limit <= x
)
return (
chain([(0, 0)], map(go, count(1)))
)
```

1. --------------------------TEST---------------------------
2. main :: IO ()

def main():

```   Tests
```
```   print('First 61 terms:')
print(showList(
take(61)(
map(fusc, count())
)
))
```
```   print('\nFirst term of each decimal magnitude:')
print('(Index, Term):')
ixs = firstFuscOfEachMagnitude()
for _ in range(0, 5):
print(next(ixs))
```

1. -------------------------GENERIC-------------------------
1. iterate :: (a -> a) -> a -> Gen [a]

def iterate(f):

```   An infinite list of repeated
applications of f to x.

def go(x):
v = x
while True:
yield v
v = f(v)
return lambda x: go(x)
```

1. showList :: [a] -> String

def showList(xs):

```   Compact stringification of a list.
return '[' + ','.join(repr(x) for x in xs) + ']'
```

1. take :: Int -> [a] -> [a]
2. 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, tuple))
else list(islice(xs, n))
)
```

1. MAIN ---

if __name__ == '__main__':

```   main()</lang>
```
Output:
```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]

First term of each decimal magnitude:
(Index, Term):
(0, 0)
(37, 11)
(1173, 108)
(35499, 1076)
(699051, 10946)```

## Racket

<lang 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)))</lang>
```
Output:
```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
```

## Raku

(formerly Perl 6)

Works with: Rakudo version 2018.12

<lang perl6>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
```

}</lang>

Output:
```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
```

## REXX

<lang rexx>/*REXX program calculates and displays the fusc (or Stern's Diatomic) sequence. */ parse arg st # xw . /*obtain optional arguments from the CL*/ if st== | st=="," then st= 0 /*Not specified? Then use the default.*/ if #== | #=="," then #= 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 #                                 /*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>=st  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 _</lang>

output   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
```
output   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

<lang ring>

1. 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
```

</lang>

Output:
```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: <lang ruby>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 } </lang>

Output:
```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

<lang ruby>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)
```

}</lang>

Output:
```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
```

## Swift

<lang swift>struct FuscSeq: Sequence, IteratorProtocol {

``` private var arr = [0, 1]
private var i = 0
```
``` mutating func next() -> Int? {
defer {
i += 1
}
```
```   guard i > 1 else {
return arr[i]
}
```
```   switch i & 1 {
case 0:
arr.append(arr[i / 2])
case 1:
arr.append(arr[(i - 1) / 2] + arr[(i + 1) / 2])
case _:
fatalError()
}
```
```   return arr.last!
}
```

}

let first = FuscSeq().prefix(61)

print("First 61: \(Array(first))")

var max = -1

for (i, n) in FuscSeq().prefix(20_000_000).enumerated() {

``` let f = String(n).count
```
``` if f > max {
max = f
```
```   print("New max: \(i): \(n)")
}
```

}</lang>

Output:
```First 61: [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]
New max: 0: 0
New max: 37: 11
New max: 1173: 108
New max: 35499: 1076
New max: 699051: 10946
New max: 19573419: 103682```

## Vala

Translation of: Nim

<lang vala>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);
```

}

void main() {

``` print("The first 61 fusc numbers:\n");
for (int i = 0; i < 61; i++)
print(@"\$(fusc(i)) ");
print("\n\nThe fusc numbers whose lengths are greater than those of previous fusc numbers:\n");
print("        n   fusc(n)\n");
print("-------------------\n");
var max_length = 0;
for (int i = 0; i < 700000; i++) {
var f = fusc(i);
var length = f.to_string().length;
if (length > max_length) {
max_length = length;
print("%9d %9d\n", i, f);
}
}
```

}</lang>

Output:
```The 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

The fusc numbers whose lengths are greater than those of previous fusc numbers:
n   fusc(n)
-------------------
0         0
37        11
1173       108
35499      1076
699051     10946
```

## Visual Basic .NET

Translation of: C#

<lang vbnet>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 </lang>

Output:
```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

<lang 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 }
```

}</lang>

Output:
```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
```