# Yellowstone sequence

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

The Yellowstone sequence, also called the Yellowstone permutation, is defined as:

For n <= 3,

```   a(n) = n
```

For n >= 4,

```   a(n) = the smallest number not already in sequence such that a(n) is relatively prime to a(n-1) and
is not relatively prime to a(n-2).
```

The sequence is a permutation of the natural numbers, and gets its name from what its authors felt was a spiking, geyser like appearance of a plot of the sequence.

Example

a(4) is 4 because 4 is the smallest number following 1, 2, 3 in the sequence that is relatively prime to the entry before it (3), and is not relatively prime to the number two entries before it (2).

Find and show as output the first  30  Yellowstone numbers.

Extra
Demonstrate how to plot, with x = n and y coordinate a(n), the first 100 Yellowstone numbers.

## 11l

Translation of: C++
```T YellowstoneGenerator
min_ = 1
n_ = 0
n1_ = 0
n2_ = 0
Set[Int] sequence_

F next()
.n2_ = .n1_
.n1_ = .n_
I .n_ < 3
.n_++
E
.n_ = .min_
L !(.n_ !C .sequence_ & gcd(.n1_, .n_) == 1 & gcd(.n2_, .n_) > 1)
.n_++
L
I .min_ !C .sequence_
L.break
.sequence_.remove(.min_)
.min_++
R .n_

print(‘First 30 Yellowstone numbers:’)
V ygen = YellowstoneGenerator()
print(ygen.next(), end' ‘’)
L(i) 1 .< 30
print(‘ ’ygen.next(), end' ‘’)
print()```
Output:
```First 30 Yellowstone numbers:
1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17
```
Translation of: Ruby
```F yellow(n)
V a = [1, 2, 3]
V b = Set([1, 2, 3])
V i = 4
L n > a.len
I i !C b & gcd(i, a.last) == 1 & gcd(i, a[(len)-2]) > 1
a.append(i)
i = 4
i++
R a

print(yellow(30))```
Output:
```[1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17]
```

## Action!

```BYTE FUNC Gcd(BYTE a,b)
BYTE tmp

IF a<b THEN
tmp=a a=b b=tmp
FI

WHILE b#0
DO
tmp=a MOD b
a=b b=tmp
OD
RETURN (a)

BYTE FUNC Contains(BYTE ARRAY a BYTE len,value)
BYTE i

FOR i=0 TO len-1
DO
IF a(i)=value THEN
RETURN (1)
FI
OD
RETURN (0)

PROC Generate(BYTE ARRAY seq BYTE count)
BYTE i,x

seq(0)=1 seq(1)=2 seq(2)=3
FOR i=3 TO COUNT-1
DO
x=1
DO
IF Contains(seq,i,x)=0 AND
Gcd(x,seq(i-1))=1 AND Gcd(x,seq(i-2))>1 THEN
EXIT
FI
x==+1
OD
seq(i)=x
OD
RETURN

PROC Main()
DEFINE COUNT="30"
BYTE ARRAY seq(COUNT)
BYTE i

Generate(seq,COUNT)
PrintF("First %B Yellowstone numbers:%E",COUNT)
FOR i=0 TO COUNT-1
DO
PrintB(seq(i)) Put(32)
OD
RETURN```
Output:
```First 30 Yellowstone numbers:
1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17
```

Translation of: C++
```with Ada.Text_IO;

procedure Yellowstone_Sequence is

generic  --  Allow more than one generator, but must be instantiated
package Yellowstones is
function Next return Integer;
function GCD (Left, Right : Integer) return Integer;
end Yellowstones;

package body Yellowstones
is
package Sequences is

--  Internal package state
N_0 : Integer := 0;
N_1 : Integer := 0;
N_2 : Integer := 0;
Seq : Sequences.Set;
Min : Integer := 1;

function GCD (Left, Right : Integer) return Integer
is (if Right = 0
then Left
else GCD (Right, Left mod Right));

function Next return Integer is
begin
N_2 := N_1;
N_1 := N_0;
if N_0 < 3 then
N_0 := N_0 + 1;
else
N_0 := Min;
while
not (not Seq.Contains (N_0)
and then GCD (N_1, N_0) = 1
and then GCD (N_2, N_0) > 1)
loop
N_0 := N_0 + 1;
end loop;
end if;
Seq.Insert (N_0);
while Seq.Contains (Min) loop
Seq.Delete (Min);
Min := Min + 1;
end loop;
return N_0;
end Next;

end Yellowstones;

procedure First_30 is
package Yellowstone is new Yellowstones;  --  New generator instance
begin
Put_Line ("First 30 Yellowstone numbers:");
for A in 1 .. 30 loop
Put (Yellowstone.Next'Image); Put (" ");
end loop;
New_Line;
end First_30;

begin
First_30;
end Yellowstone_Sequence;
```
Output:
```First 30 Yellowstone numbers:
1  2  3  4  9  8  15  14  5  6  25  12  35  16  7  10  21  20  27  22  39  11  13  33  26  45  28  51  32  17
```

## ALGOL 68

```BEGIN # find members of the yellowstone sequence: starting from 1, 2, 3 the   #
# subsequent members are the lowest number coprime to the previous one  #
# and not coprime to the one before that, that haven't appeared in the  #
# sequence yet                                                          #
# iterative Greatest Common Divisor routine, returns the gcd of m and n   #
PROC gcd = ( INT m, n )INT:
BEGIN
INT a := ABS m, b := ABS n;
WHILE b /= 0 DO
INT new a = b;
b        := a MOD b;
a        := new a
OD;
a
END # gcd # ;
# returns an array of the Yellowstone seuence up to n                     #
OP   YELLOWSTONE = ( INT n )[]INT:
BEGIN
[ 1 : n ]INT result;
IF n > 0 THEN
result[ 1 ] := 1;
IF n > 1 THEN
result[ 2 ] := 2;
IF n > 2 THEN
result[ 3 ] := 3;
# guess the maximum element will be n, if it is larger, used will be enlarged #
REF[]BOOL used := HEAP[ 1 : n ]BOOL;
used[ 1 ] := used[ 2 ] := used[ 3 ] := TRUE;
FOR i FROM 4 TO UPB used DO used[ i ] := FALSE OD;
FOR i FROM 4 TO UPB result DO
INT p1      = result[ i - 1 ];
INT p2      = result[ i - 2 ];
BOOL found := FALSE;
IF j > UPB used THEN
# not enough elements in used - enlarge it #
REF[]BOOL new used := HEAP[ 1 : 2 * UPB used ]BOOL;
new used[ 1 : UPB used ] := used;
FOR k FROM UPB used + 1 TO UPB new used DO new used[ k ] := FALSE OD;
used := new used
FI;
IF NOT used[ j ] THEN
IF found := gcd( j, p1 ) = 1 AND gcd( j, p2 ) /= 1
THEN
result[ i ] := j;
used[   j ] := TRUE
FI
FI
OD
OD
FI
FI
FI;
result
END # YELLOWSTONE # ;
[]INT ys = YELLOWSTONE 30;
FOR i TO UPB ys DO
print( ( " ", whole( ys[ i ], 0 ) ) )
OD
END```
Output:
``` 1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17
```

## Arturo

```yellowstone: function [n][
result: new [1 2 3]
present: new [1 2 3]
start: new 4
while [n > size result][
candidate: new start
while ø [
if all? @[
not? contains? present candidate
1 = gcd @[candidate last result]
1 <> gcd @[candidate get result (size result)-2]
][
'result ++ candidate
'present ++ candidate
while [contains? present start] -> inc 'start
break
]
inc 'candidate
]
]
return result
]

print yellowstone 30
```
Output:
`1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17`

## AutoHotkey

```A := [], in_seq := []
loop 30 {
n := A_Index
if n <=3
A[n] := n,    in_seq[n] := true
else while true
{
s := A_Index
if !in_seq[s] && relatively_prime(s, A[n-1]) && !relatively_prime(s, A[n-2])
{
A[n] := s
in_seq[s] := true
break
}
}
}
for i, v in A
result .= v ","
MsgBox % result := "[" Trim(result, ",") "]"
return
;--------------------------------------
relatively_prime(a, b){
return (GCD(a, b) = 1)
}
;--------------------------------------
GCD(a, b) {
while b
b := Mod(a | 0x0, a := b)
return a
}
```
Output:
`[1,2,3,4,9,8,15,14,5,6,25,12,35,16,7,10,21,20,27,22,39,11,13,33,26,45,28,51,32,17]`

## C

Translation of: Ruby
```#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>

typedef struct lnode_t {
struct lnode_t *prev;
struct lnode_t *next;
int v;
} Lnode;

Lnode *make_list_node(int v) {
Lnode *node = malloc(sizeof(Lnode));
if (node == NULL) {
return NULL;
}
node->v = v;
node->prev = NULL;
node->next = NULL;
return node;
}

void free_lnode(Lnode *node) {
if (node == NULL) {
return;
}

node->v = 0;
node->prev = NULL;
free_lnode(node->next);
node->next = NULL;
}

typedef struct list_t {
Lnode *front;
Lnode *back;
size_t len;
} List;

List *make_list() {
List *list = malloc(sizeof(List));
if (list == NULL) {
return NULL;
}
list->front = NULL;
list->back = NULL;
list->len = 0;
return list;
}

void free_list(List *list) {
if (list == NULL) {
return;
}
list->len = 0;
list->back = NULL;
free_lnode(list->front);
list->front = NULL;
}

void list_insert(List *list, int v) {
Lnode *node;

if (list == NULL) {
return;
}

node = make_list_node(v);
if (list->front == NULL) {
list->front = node;
list->back = node;
list->len = 1;
} else {
node->prev = list->back;
list->back->next = node;
list->back = node;
list->len++;
}
}

void list_print(List *list) {
Lnode *it;

if (list == NULL) {
return;
}

for (it = list->front; it != NULL; it = it->next) {
printf("%d ", it->v);
}
}

int list_get(List *list, int idx) {
Lnode *it = NULL;

if (list != NULL && list->front != NULL) {
int i;
if (idx < 0) {
it = list->back;
i = -1;
while (it != NULL && i > idx) {
it = it->prev;
i--;
}
} else {
it = list->front;
i = 0;
while (it != NULL && i < idx) {
it = it->next;
i++;
}
}
}

if (it == NULL) {
return INT_MIN;
}
return it->v;
}

///////////////////////////////////////

typedef struct mnode_t {
int k;
bool v;
struct mnode_t *next;
} Mnode;

Mnode *make_map_node(int k, bool v) {
Mnode *node = malloc(sizeof(Mnode));
if (node == NULL) {
return node;
}
node->k = k;
node->v = v;
node->next = NULL;
return node;
}

void free_mnode(Mnode *node) {
if (node == NULL) {
return;
}
node->k = 0;
node->v = false;
free_mnode(node->next);
node->next = NULL;
}

typedef struct map_t {
Mnode *front;
} Map;

Map *make_map() {
Map *map = malloc(sizeof(Map));
if (map == NULL) {
return NULL;
}
map->front = NULL;
return map;
}

void free_map(Map *map) {
if (map == NULL) {
return;
}
free_mnode(map->front);
map->front = NULL;
}

void map_insert(Map *map, int k, bool v) {
if (map == NULL) {
return;
}
if (map->front == NULL) {
map->front = make_map_node(k, v);
} else {
Mnode *it = map->front;
while (it->next != NULL) {
it = it->next;
}
it->next = make_map_node(k, v);
}
}

bool map_get(Map *map, int k) {
if (map != NULL) {
Mnode *it = map->front;
while (it != NULL && it->k != k) {
it = it->next;
}
if (it != NULL) {
return it->v;
}
}
return false;
}

///////////////////////////////////////

int gcd(int u, int v) {
if (u < 0) u = -u;
if (v < 0) v = -v;
if (v) {
while ((u %= v) && (v %= u));
}
return u + v;
}

List *yellow(size_t n) {
List *a;
Map *b;
int i;

a = make_list();
list_insert(a, 1);
list_insert(a, 2);
list_insert(a, 3);

b = make_map();
map_insert(b, 1, true);
map_insert(b, 2, true);
map_insert(b, 3, true);

i = 4;

while (n > a->len) {
if (!map_get(b, i) && gcd(i, list_get(a, -1)) == 1 && gcd(i, list_get(a, -2)) > 1) {
list_insert(a, i);
map_insert(b, i, true);
i = 4;
}
i++;
}

free_map(b);
return a;
}

int main() {
List *a = yellow(30);
list_print(a);
free_list(a);
putc('\n', stdout);
return 0;
}
```
Output:
`1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17`

## C++

```#include <iostream>
#include <numeric>
#include <set>

template <typename integer>
class yellowstone_generator {
public:
integer next() {
n2_ = n1_;
n1_ = n_;
if (n_ < 3) {
++n_;
} else {
for (n_ = min_; !(sequence_.count(n_) == 0
&& std::gcd(n1_, n_) == 1
&& std::gcd(n2_, n_) > 1); ++n_) {}
}
sequence_.insert(n_);
for (;;) {
auto it = sequence_.find(min_);
if (it == sequence_.end())
break;
sequence_.erase(it);
++min_;
}
return n_;
}
private:
std::set<integer> sequence_;
integer min_ = 1;
integer n_ = 0;
integer n1_ = 0;
integer n2_ = 0;
};

int main() {
std::cout << "First 30 Yellowstone numbers:\n";
yellowstone_generator<unsigned int> ygen;
std::cout << ygen.next();
for (int i = 1; i < 30; ++i)
std::cout << ' ' << ygen.next();
std::cout << '\n';
return 0;
}
```
Output:
```First 30 Yellowstone numbers:
1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17
```

## D

Translation of: C++
```import std.numeric;
import std.range;
import std.stdio;

class Yellowstone {
private bool[int] sequence_;
private int min_ = 1;
private int n_ = 0;
private int n1_ = 0;
private int n2_ = 0;

public this() {
popFront();
}

public bool empty() {
return false;
}

public int front() {
return n_;
}

public void popFront() {
n2_ = n1_;
n1_ = n_;
if (n_ < 3) {
++n_;
} else {
for (n_ = min_;
!(n_ !in sequence_ && gcd(n1_, n_) == 1 && gcd(n2_, n_) > 1);
++n_) {
// empty
}
}
sequence_[n_] = true;
while (true) {
if (min_ !in sequence_) {
break;
}
sequence_.remove(min_);
++min_;
}
}
}

void main() {
new Yellowstone().take(30).writeln();
}
```
Output:
`[1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17]`

## Delphi

Translation of: Go

Boost.Generics.Collection and Boost.Process are part of DelphiBoostLib.

```program Yellowstone_sequence;

{\$APPTYPE CONSOLE}

uses
System.SysUtils,
Boost.Generics.Collection,
Boost.Process;

function gdc(x, y: Integer): Integer;
begin
while y <> 0 do
begin
var tmp := x;
x := y;
y := tmp mod y;
end;
Result := x;
end;

function Yellowstone(n: Integer): TArray<Integer>;
var
m: TDictionary<Integer, Boolean>;
a: TArray<Integer>;
begin
m.Init;
SetLength(a, n + 1);
for var i := 1 to 3 do
begin
a[i] := i;
m[i] := True;
end;

var min := 4;

for var c := 4 to n do
begin
var i := min;
repeat
if not m[i, false] and (gdc(a[c - 1], i) = 1) and (gdc(a[c - 2], i) > 1) then
begin
a[c] := i;
m[i] := true;
if i = min then
inc(min);
Break;
end;
inc(i);
until false;
end;

Result := copy(a, 1, length(a));
end;

begin
var x: TArray<Integer>;
SetLength(x, 100);
for var i in Range(100) do
x[i] := i + 1;

var y := yellowstone(High(x));

writeln('The first 30 Yellowstone numbers are:');
for var i := 0 to 29 do
Write(y[i], ' ');
Writeln;

//Plotting

var plot := TPipe.Create('gnuplot -p', True);
plot.WritelnA('unset key; plot ''-''');

for var i := 0 to High(x) do
plot.WriteA('%d %d'#10, [x[i], y[i]]);
plot.WritelnA('e');

writeln('Press enter to close');
plot.Kill;
plot.Free;
end.
```
Output:
```The first 30 Yellowstone numbers are:
1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17
Press enter to close```

## EasyLang

Translation of: Lua
```func gcd a b .
if b = 0
return a
.
return gcd b (a mod b)
.
proc remove_at i . a[] .
for j = i + 1 to len a[]
a[j - 1] = a[j]
.
len a[] -1
.
proc yellowstone count . yellow[] .
yellow[] = [ 1 2 3 ]
num = 4
while len yellow[] < count
yell1 = yellow[len yellow[] - 1]
yell2 = yellow[len yellow[]]
for i to len notyellow[]
test = notyellow[i]
if gcd yell1 test > 1 and gcd yell2 test = 1
break 1
.
.
if i <= len notyellow[]
yellow[] &= notyellow[i]
remove_at i notyellow[]
else
while gcd yell1 num <= 1 or gcd yell2 num <> 1
notyellow[] &= num
num += 1
.
yellow[] &= num
num += 1
.
.
.
print "First 30 values in the yellowstone sequence:"
yellowstone 30 yellow[]
print yellow[]```

## Factor

Works with: Factor version 0.99 2020-01-23
```USING: accessors assocs colors.constants
combinators.short-circuit io kernel math prettyprint sequences

: yellowstone? ( n hs seq -- ? )
{
[ drop in? not ]
[ nip last gcd nip 1 = ]
[ nip dup length 2 - swap nth gcd nip 1 > ]
} 3&& ;

: next-yellowstone ( hs seq -- n )
[ 4 ] 2dip [ 3dup yellowstone? ] [ [ 1 + ] 2dip ] until
2drop ;

: next ( hs seq -- hs' seq' )
2dup next-yellowstone [ suffix! ] [ pick adjoin ] bi ;

: <yellowstone> ( n -- seq )
[ HS{ 1 2 3 } clone dup V{ } set-like ] dip dup 3 <=
[ head nip ] [ 3 - [ next ] times nip ] if ;

! Show first 30 Yellowstone numbers.

"First 30 Yellowstone numbers:" print
30 <yellowstone> [ pprint bl ] each nl

! Plot first 100 Yellowstone numbers.

chart new { { 0 100 } { 0 175 } } >>axes
line new COLOR: blue >>color
100 <iota> 100 <yellowstone> zip >>data
```
Output:
```First 30 Yellowstone numbers:
1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17
```

## Forth

```: array create cells allot ;
: th cells + ;                         \ some helper words

30 constant #yellow                    \ number of yellowstones

#yellow array y                        \ create array
( n1 n2 -- n3)
: gcd dup if tuck mod recurse exit then drop ;
: init 3 0 do i 1+ y i th ! loop ;     ( --)
: show cr #yellow 0 do y i th ? loop ; ( --)
: gcd-y[] - cells y + @ over gcd ;     ( k i n -- k gcd )
: loop1 begin 1+ over 2 gcd-y[] 1 = >r over 1 gcd-y[] 1 > r> or 0= until ;
: loop2 over true swap 0 ?do over y i th @ = if 0= leave then loop ;
: yellow #yellow 3 do i 3 begin loop1 loop2 until y rot th ! loop ;
: main init yellow show ;

main
```
Output:
```main
1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17  ok```

## FreeBASIC

```function gcd(a as uinteger, b as uinteger) as uinteger
if b = 0 then return a
return gcd( b, a mod b )
end function

dim as uinteger i, j, k, Y(1 to 100)

Y(1) = 1 : Y(2) = 2: Y(3) = 3

for i = 4 to 100
k = 3
print i
do
k += 1
if gcd( k, Y(i-2) ) = 1 orelse gcd( k, Y(i-1) ) > 1 then continue do
for j = 1 to i-1
if Y(j)=k then continue do
next j
Y(i) = k
exit do
loop
next i

for i = 1 to 30
print str(Y(i))+" ";
next i
print
screen 13
for i = 1 to 100
pset (i, 200-Y(i)), 31
next i

while inkey=""
wend
end```
Output:
`1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17`

## Go

This uses Gnuplot-X11 to do the plotting rather than a third party Go plotting library.

```package main

import (
"fmt"
"log"
"os/exec"
)

func gcd(x, y int) int {
for y != 0 {
x, y = y, x%y
}
return x
}

func yellowstone(n int) []int {
m := make(map[int]bool)
a := make([]int, n+1)
for i := 1; i < 4; i++ {
a[i] = i
m[i] = true
}
min := 4
for c := 4; c <= n; c++ {
for i := min; ; i++ {
if !m[i] && gcd(a[c-1], i) == 1 && gcd(a[c-2], i) > 1 {
a[c] = i
m[i] = true
if i == min {
min++
}
break
}
}
}
return a[1:]
}

func check(err error) {
if err != nil {
log.Fatal(err)
}
}

func main() {
x := make([]int, 100)
for i := 0; i < 100; i++ {
x[i] = i + 1
}
y := yellowstone(100)
fmt.Println("The first 30 Yellowstone numbers are:")
fmt.Println(y[:30])
g := exec.Command("gnuplot", "-persist")
w, err := g.StdinPipe()
check(err)
check(g.Start())
fmt.Fprintln(w, "unset key; plot '-'")
for i, xi := range x {
fmt.Fprintf(w, "%d %d\n", xi, y[i])
}
fmt.Fprintln(w, "e")
w.Close()
g.Wait()
}
```
Output:
```The first 30 Yellowstone numbers are:
[1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17]
```

```import Data.List (unfoldr)

yellowstone :: [Integer]
yellowstone = 1 : 2 : 3 : unfoldr (Just . f) (2, 3, [4 ..])
where
f ::
(Integer, Integer, [Integer]) ->
(Integer, (Integer, Integer, [Integer]))
f (p2, p1, rest) = (next, (p1, next, rest_))
where
(next, rest_) = select rest
select :: [Integer] -> (Integer, [Integer])
select (x : xs)
| gcd x p1 == 1 && gcd x p2 /= 1 = (x, xs)
| otherwise = (y, x : ys)
where
(y, ys) = select xs

main :: IO ()
main = print \$ take 30 yellowstone
```
Output:
`[1,2,3,4,9,8,15,14,5,6,25,12,35,16,7,10,21,20,27,22,39,11,13,33,26,45,28,51,32,17]`

Or, defining the Yellowstone permutation in terms of iterate, rather than unfoldr,

and displaying a chart of the first 100 terms:

```import Codec.Picture
import Data.Bifunctor (second)
import Diagrams.Backend.Rasterific
import Diagrams.Prelude
import Graphics.Rendering.Chart.Backend.Diagrams
import Graphics.Rendering.Chart.Easy

----------------- YELLOWSTONE PERMUTATION ----------------
yellowstone :: [Integer]
yellowstone =
1 :
2 :
(active <\$> iterate nextWindow (2, 3, [4 ..]))
where
nextWindow (p2, p1, rest) = (p1, n, residue)
where
[rp2, rp1] = relativelyPrime <\$> [p2, p1]
go (x : xs)
| rp1 x && not (rp2 x) = (x, xs)
| otherwise = second ((:) x) (go xs)
(n, residue) = go rest
active (_, x, _) = x

relativelyPrime :: Integer -> Integer -> Bool
relativelyPrime a b = 1 == gcd a b

---------- 30 FIRST TERMS, AND CHART OF FIRST 100 --------
main :: IO (Image PixelRGBA8)
main = do
print \$ take 30 yellowstone
env <- chartEnv
return \$
chartRender env \$
plot
( line
"Yellowstone terms"
[zip [1 ..] (take 100 yellowstone)]
)

--------------------- CHART GENERATION -------------------
chartRender ::
(Default r, ToRenderable r) =>
DEnv Double ->
EC r () ->
Image PixelRGBA8
chartRender env ec =
renderDia
Rasterific
( RasterificOptions
(mkWidth (fst (envOutputSize env)))
)
\$ fst \$ runBackendR env (toRenderable (execEC ec))

------------------------ LOCAL FONT ----------------------
chartEnv :: IO (DEnv Double)
chartEnv = do
let fontChosen fs =
case ( _font_name fs,
_font_slant fs,
_font_weight fs
) of
( "sans-serif",
FontSlantNormal,
FontWeightNormal
) -> sansR
( "sans-serif",
FontSlantNormal,
FontWeightBold
) -> sansRB
return \$ createEnv vectorAlignmentFns 640 400 fontChosen
```
Output:
`[1,2,3,4,9,8,15,14,5,6,25,12,35,16,7,10,21,20,27,22,39,11,13,33,26,45,28,51,32,17]`

## J

### tacit

```Until=: 2 :'u^:(0-:v)^:_'
assert 44 -: >:Until(>&43) 32  NB. increment until exceeding 43
gcd=: +.
coprime=: 1 = gcd
condition=: 0 1 -: (coprime _2&{.)   NB. trial coprime most recent 2,  nay and yay
append=: ,      NB. concatenate
novel=: -.@e.   NB. x is not a member of y
term=: >:@:]Until((condition *. novel)~) 4:
ys=: (append term)@]^:(0 >. _3+[) prepare
assert (ys 30) -: 1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17
```

### explicit

```GCD=: +.
relatively_prime=: 1 = GCD

yellowstone=: {{
s=. 1 2 3            NB. initial sequence
while. y > # s do.
z=. <./(1+s)-.s    NB. lowest positive inteeger not in sequence
while. if. 0 1 -: z relatively_prime _2{.s do. z e. s end. do.
z=. z+1
end.   NB. find next value for sequence
s=. s, z
end.
}}
```
```   yellowstone 30
1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17

'marker'plot yellowstone 100
```

## Java

```import java.util.ArrayList;
import java.util.List;

public class YellowstoneSequence {

public static void main(String[] args) {
System.out.printf("First 30 values in the yellowstone sequence:%n%s%n", yellowstoneSequence(30));
}

private static List<Integer> yellowstoneSequence(int sequenceCount) {
List<Integer> yellowstoneList = new ArrayList<Integer>();
int num = 4;
List<Integer> notYellowstoneList = new ArrayList<Integer>();
int yellowSize = 3;
while ( yellowSize < sequenceCount ) {
int found = -1;
for ( int index = 0 ; index < notYellowstoneList.size() ; index++ ) {
int test = notYellowstoneList.get(index);
if ( gcd(yellowstoneList.get(yellowSize-2), test) > 1 && gcd(yellowstoneList.get(yellowSize-1), test) == 1 ) {
found = index;
break;
}
}
if ( found >= 0 ) {
yellowSize++;
}
else {
while ( true ) {
if ( gcd(yellowstoneList.get(yellowSize-2), num) > 1 && gcd(yellowstoneList.get(yellowSize-1), num) == 1 ) {
yellowSize++;
num++;
break;
}
num++;
}
}
}
return yellowstoneList;
}

private static final int gcd(int a, int b) {
if ( b == 0 ) {
return a;
}
return gcd(b, a%b);
}

}
```
Output:
```First 30 values in the yellowstone sequence:
[1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17]
```

## JavaScript

Translation of: Python
Works with: ES6
```(() => {
'use strict';

// yellowstone :: Generator [Int]
function* yellowstone() {
// A non finite stream of terms in the
// Yellowstone permutation of the natural numbers.
// OEIS A098550
const nextWindow = ([p2, p1, rest]) => {
const [rp2, rp1] = [p2, p1].map(
relativelyPrime
);
const go = xxs => {
const [x, xs] = Array.from(
uncons(xxs).Just
);
return rp1(x) && !rp2(x) ? (
Tuple(x)(xs)
) : secondArrow(cons(x))(
go(xs)
);
};
return [p1, ...Array.from(go(rest))];
};
const A098550 = fmapGen(x => x[1])(
iterate(nextWindow)(
[2, 3, enumFrom(4)]
)
);
yield 1
yield 2
while (true)(
yield A098550.next().value
)
};

// relativelyPrime :: Int -> Int -> Bool
const relativelyPrime = a =>
// True if a is relatively prime to b.
b => 1 === gcd(a)(b);

// ------------------------TEST------------------------
const main = () => console.log(
take(30)(
yellowstone()
)
);

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

// Just :: a -> Maybe a
const Just = x => ({
type: 'Maybe',
Nothing: false,
Just: x
});

// Nothing :: Maybe a
const Nothing = () => ({
type: 'Maybe',
Nothing: true,
});

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

// abs :: Num -> Num
const abs =
// Absolute value of a given number - without the sign.
Math.abs;

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

// enumFrom :: Enum a => a -> [a]
function* enumFrom(x) {
// A non-finite succession of enumerable
// values, starting with the value x.
let v = x;
while (true) {
yield v;
v = 1 + v;
}
}

// fmapGen <\$> :: (a -> b) -> Gen [a] -> Gen [b]
const fmapGen = f =>
function*(gen) {
let v = take(1)(gen);
while (0 < v.length) {
yield(f(v[0]))
v = take(1)(gen)
}
};

// gcd :: Int -> Int -> Int
const gcd = x => y => {
const
_gcd = (a, b) => (0 === b ? a : _gcd(b, a % b)),
abs = Math.abs;
return _gcd(abs(x), abs(y));
};

// iterate :: (a -> a) -> a -> Gen [a]
const iterate = f =>
function*(x) {
let v = x;
while (true) {
yield(v);
v = f(v);
}
};

// length :: [a] -> Int
const length = xs =>
// Returns Infinity over objects without finite
// length. This enables zip and zipWith to choose
// the shorter argument when one is non-finite,
// like cycle, repeat etc
(Array.isArray(xs) || 'string' === typeof xs) ? (
xs.length
) : Infinity;

// secondArrow :: (a -> b) -> ((c, a) -> (c, b))
const secondArrow = f => xy =>
// A function over a simple value lifted
// to a function over a tuple.
// f (a, b) -> (a, f(b))
Tuple(xy[0])(
f(xy[1])
);

// take :: Int -> [a] -> [a]
// take :: Int -> String -> String
const take = n =>
// The first n elements of a list,
// string of characters, or stream.
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];
}));

// uncons :: [a] -> Maybe (a, [a])
const uncons = xs => {
// Just a tuple of the head of xs and its tail,
// Or Nothing if xs is an empty list.
const lng = length(xs);
return (0 < lng) ? (
Infinity > lng ? (
Just(Tuple(xs[0])(xs.slice(1))) // Finite list
) : (() => {
const nxt = take(1)(xs);
return 0 < nxt.length ? (
Just(Tuple(nxt[0])(xs))
) : Nothing();
})() // Lazy generator
) : Nothing();
};

// MAIN ---
return main();
})();
```
Output:
`1,2,3,4,9,8,15,14,5,6,25,12,35,16,7,10,21,20,27,22,39,11,13,33,26,45,28,51,32,17`

## jq

Works with: jq
```# jq optimizes the recursive call of _gcd in the following:
def gcd(a;b):
def _gcd:
if .[1] != 0 then [.[1], .[0] % .[1]] | _gcd else .[0] end;
[a,b] | _gcd ;

# emit the yellowstone sequence as a stream
def yellowstone:
1,2,3,
({ a: [2, 3],                               # the last two items only
b: {"1":  true, "2": true, "3" : true},  # a record, to avoid having to save the entire history
start: 4 }
| foreach range(1; infinite) as \$n (.;
first(
.b as \$b
| .start = first( range(.start;infinite) | select(\$b[tostring]|not) )
| foreach range(.start; infinite) as \$i (.;
.emit = null
| (\$i|tostring) as \$is
| if .b[\$is] then .
# "a(n) is relatively prime to a(n-1) and is not relatively prime to a(n-2)"
elif (gcd(\$i; .a[1]) == 1) and (gcd(\$i; .a[0]) > 1)
then .emit = \$i
| .a = [.a[1], \$i]
| .b[\$is] = true
else .
end;
select(.emit)) );
.emit ));```

```"The first 30 entries of the Yellowstone permutation:",
[limit(30;yellowstone)]```
Output:
```The first 30 entries of the Yellowstone permutation:
[1,2,3,4,9,8,15,14,5,6,25,12,35,16,7,10,21,20,27,22,39,11,13,33,26,45,28,51,32,17]
```

## Julia

```using Plots

function yellowstone(N)
a = [1, 2, 3]
b = Dict(1 => 1, 2 => 1, 3 => 1)
start = 4
while length(a) < N
inseries = true
for i in start:typemax(Int)
if inseries
start += 1
end
else
inseries = false
end
if !haskey(b, i) && (gcd(i, a[end]) == 1) && (gcd(i, a[end - 1]) > 1)
push!(a, i)
b[i] = 1
break
end
end
end
return a
end

println("The first 30 entries of the Yellowstone permutation:\n", yellowstone(30))

x = 1:100
y = yellowstone(100)
plot(x, y)
```
Output:
```The first 30 entries of the Yellowstone permutation:
[1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17]
```

## Kotlin

Translation of: Java
```fun main() {
println("First 30 values in the yellowstone sequence:")
println(yellowstoneSequence(30))
}

private fun yellowstoneSequence(sequenceCount: Int): List<Int> {
val yellowstoneList = mutableListOf(1, 2, 3)
var num = 4
val notYellowstoneList = mutableListOf<Int>()
var yellowSize = 3
while (yellowSize < sequenceCount) {
var found = -1
for (index in notYellowstoneList.indices) {
val test = notYellowstoneList[index]
if (gcd(yellowstoneList[yellowSize - 2], test) > 1 && gcd(
yellowstoneList[yellowSize - 1], test
) == 1
) {
found = index
break
}
}
if (found >= 0) {
yellowSize++
} else {
while (true) {
if (gcd(yellowstoneList[yellowSize - 2], num) > 1 && gcd(
yellowstoneList[yellowSize - 1], num
) == 1
) {
yellowSize++
num++
break
}
num++
}
}
}
return yellowstoneList
}

private fun gcd(a: Int, b: Int): Int {
return if (b == 0) {
a
} else gcd(b, a % b)
}
```
Output:
```First 30 values in the yellowstone sequence:
[1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17]```

## Lua

Translation of: Java
```function gcd(a, b)
if b == 0 then
return a
end
return gcd(b, a % b)
end

function printArray(a)
io.write('[')
for i,v in pairs(a) do
if i > 1 then
io.write(', ')
end
io.write(v)
end
io.write(']')
return nil
end

function removeAt(a, i)
local na = {}
for j,v in pairs(a) do
if j ~= i then
table.insert(na, v)
end
end
return na
end

function yellowstone(sequenceCount)
local yellow = {1, 2, 3}
local num = 4
local notYellow = {}
local yellowSize = 3
while yellowSize < sequenceCount do
local found = -1
for i,test in pairs(notYellow) do
if gcd(yellow[yellowSize - 1], test) > 1 and gcd(yellow[yellowSize - 0], test) == 1 then
found = i
break
end
end
if found >= 0 then
table.insert(yellow, notYellow[found])
notYellow = removeAt(notYellow, found)
yellowSize = yellowSize + 1
else
while true do
if gcd(yellow[yellowSize - 1], num) > 1 and gcd(yellow[yellowSize - 0], num) == 1 then
table.insert(yellow, num)
yellowSize = yellowSize + 1
num = num + 1
break
end
table.insert(notYellow, num)
num = num + 1
end
end
end
return yellow
end

function main()
print("First 30 values in the yellowstone sequence:")
printArray(yellowstone(30))
print()
end

main()
```
Output:
```First 30 values in the yellowstone sequence:
[1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17]```

## Mathematica / Wolfram Language

```state = {1, 2, 3};
MakeNext[state_List] := Module[{i = First[state], done = False, out},
While[! done,
If[FreeQ[state, i],
If[GCD[Last[state], i] == 1,
If[GCD[state[[-2]], i] > 1,
out = Append[state, i];
done = True;
]
]
];
i++;
];
out
]
Nest[MakeNext, state, 30 - 3]
ListPlot[%]
```
Output:
```{1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17}
(* Graphical visualisation of the data *)```

## Nim

### Procedure version

This version uses a set and, so, is limited to 65536 elements. It is easy to change this limit by using a HashSet (standard module “sets”) instead of a set. See the iterator version which uses such a HashSet.

```import math

proc yellowstone(n: int): seq[int] =
assert n >= 3
result = @[1, 2, 3]
var present = {1, 2, 3}
var start = 4
while result.len < n:
var candidate = start
while true:
if candidate notin present and gcd(candidate, result[^1]) == 1 and gcd(candidate, result[^2]) != 1:
present.incl candidate
while start in present: inc start
break
inc candidate

echo yellowstone(30)
```
Output:
`@[1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17]`

### Iterator version

This version uses a HashSet, but using a set as in the previous version is possible if we accept the limit of 65536 elements.

```import math, sets

iterator yellowstone(n: int): int =
assert n >= 3
for i in 1..3: yield i
var present = [1, 2, 3].toHashSet
var prevLast = 2
var last = 3
var start = 4
for _ in 4..n:
var candidate = start
while true:
if candidate notin present and gcd(candidate, last) == 1 and gcd(candidate, prevLast) != 1:
yield candidate
present.incl candidate
prevLast = last
last = candidate
while start in present: inc start
break
inc candidate

for n in yellowstone(30):
stdout.write " ", n
echo()
```
Output:
` 1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17`

## PARI/GP

```yellowstone(n) = {
my(a=3, o=2, u=[]);
if(n<3, return(n)); \\ Base case: return n if it is less than 3
print1("1, 2");  \\ Print initial values

for(i = 4, n,  \\ Iterate from 4 to n
print1(", "a);  \\ Print current value of a
u = setunion(u, Set(a));  \\ Add a to the set u

\\ Remove consecutive elements from u
while(#u > 1 && u[2] == u[1] + 1,
u = vecextract(u, "^1")
);

\\ Find next value of a
for(k = u[1] + 1, 1e10,
if(gcd(k, o) <= 1, next);  \\ Skip if gcd(k, o) is greater than 1
if(setsearch(u, k), next);  \\ Skip if k is in set u
if(gcd(k, a) != 1, next);  \\ Skip if gcd(k, a) is not 1
o = a;  \\ Update o to current a
a = k;  \\ Update a to k
break
)
);

a  \\ Return the final value of a
}

yellowstone(20);  \\ Call the function with n = 20```
Output:
```1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27
```

## Perl

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

use List::Util qw(first);
use GD::Graph::bars;

use constant Inf  => 1e5;

sub gcd {
my (\$u, \$v) = @_;
while (\$v) {
(\$u, \$v) = (\$v, \$u % \$v);
}
return abs(\$u);
}

sub yellowstone {
my(\$terms) = @_;
my @s = (1, 2, 3);
my @used = (1) x 4;
my \$min  = 3;
while (1) {
my \$index = first { not defined \$used[\$_] and gcd(\$_,\$s[-2]) != 1 and gcd(\$_,\$s[-1]) == 1 } \$min .. Inf;
\$used[\$index] = 1;
\$min = (first { not defined \$used[\$_] } 0..@used-1) || @used-1;
push @s, \$index;
last if @s == \$terms;
}
@s;
}

say "The first 30 terms in the Yellowstone sequence:\n" . join ' ', yellowstone(30);

my @data = ( [1..500], [yellowstone(500)]);
my \$graph = GD::Graph::bars->new(800, 600);
\$graph->set(
title          => 'Yellowstone sequence',
y_max_value    => 1400,
x_tick_number  => 5,
r_margin       => 10,
dclrs          => [ 'blue' ],
) or die \$graph->error;
my \$gd = \$graph->plot(\@data) or die \$graph->error;

open my \$fh, '>', 'yellowstone-sequence.png';
binmode \$fh;
print \$fh \$gd->png();
close \$fh;
```
Output:
```The first 30 terms in the Yellowstone sequence:
1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17```

See graph at off-site PNG image

## Phix

Library: Phix/pGUI
Library: Phix/online

You can run this online here.

```--
-- demo\rosetta\Yellowstone_sequence.exw
--
with javascript_semantics
requires("1.0.2")

function yellowstone(integer N)
sequence a = {1, 2, 3},
b = repeat(true,3)
integer i = 4
while length(a) < N do
if (i>length(b) or b[i]=false)
and gcd(i,a[\$])=1
and gcd(i,a[\$-1])>1 then
a &= i
if i>length(b) then
b &= repeat(false,i-length(b))
end if
b[i] = true
i = 4
end if
i += 1
end while
return a
end function

printf(1,"The first 30 entries of the Yellowstone permutation:\n%v\n", {yellowstone(30)})

-- a simple plot:
include pGUI.e
include IupGraph.e

function get_data(Ihandle graph)
sequence y500 = yellowstone(500)
integer {w,h} = IupGetIntInt(graph,"DRAWSIZE")
IupSetInt(graph,"XTICK",iff(w<640?iff(h<300?100:50):20))
IupSetInt(graph,"YTICK",iff(h<250?iff(h<140?iff(h<120?700:350):200):100))
return {{tagset(500),y500,CD_RED}}
end function

IupOpen()
Ihandle graph = IupGraph(get_data,"RASTERSIZE=960x600")
IupSetAttributes(graph,`GTITLE="Yellowstone Numbers"`)
IupSetInt(graph,"TITLESTYLE",CD_ITALIC)
IupSetAttributes(graph,`XNAME="n", YNAME="a(n)"`)
IupSetAttributes(graph,"XTICK=20,XMIN=0,XMAX=500")
IupSetAttributes(graph,"YTICK=100,YMIN=0,YMAX=1400")
Ihandle dlg = IupDialog(graph,`TITLE="Yellowstone Names"`)
IupSetAttributes(dlg,"MINSIZE=290x140")
IupShow(dlg)
if platform()!=JS then
IupMainLoop()
IupClose()
end if
```
Output:
```The first 30 entries of the Yellowstone permutation:
{1,2,3,4,9,8,15,14,5,6,25,12,35,16,7,10,21,20,27,22,39,11,13,33,26,45,28,51,32,17}
```

## Phixmonti

Translation of: Ruby

Require Utilitys library version 1.3

```include ..\Utilitys.pmt

def gcd /# u v -- n #/
abs int swap abs int swap

dup
while
over over mod rot drop dup
endwhile
drop
enddef

def test enddef

def yellow var n
( 1 2 3 ) var a
newd ( 1 true ) setd ( 2 true ) setd ( 3 true ) setd var b
4 var i
test
while
b i getd "Unfound" == >ps
a -1 get >ps -2 get
i gcd 1 > ps> i gcd 1 == ps>
and and if
i 0 put var a
( i true ) setd var b
4 var i
else
drop drop
endif
i 1 + var i
test
endwhile
a
enddef

def test n a len nip > enddef

"The first 30 entries of the Yellowstone permutation:" ? 30 yellow ?```
Output:
```The first 30 entries of the Yellowstone permutation:
[1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17]

=== Press any key to exit ===```

## PicoLisp

```(load "@lib/frac.l")
(de yellow (N)
(let (L (list 3 2 1)  I 4  C 3  D)
(while (> N C)
(when
(and
(not (idx 'D I))
(=1 (gcd I (get L 1)))
(> (gcd I (get L 2)) 1) )
(push 'L I)
(idx 'D I T)
(setq I 4)
(inc 'C) )
(inc 'I) )
(flip L) ) )
(println (yellow 30))```
Output:
```(1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17)
```

## PureBasic

```Procedure.i gcd(x.i,y.i)
While y<>0 : t=x : x=y : y=t%y : Wend : ProcedureReturn x
EndProcedure

If OpenConsole()
Dim Y.i(100)
For i=1 To 100
If i<=3 : Y(i)=i : Continue : EndIf : k=3
Repeat
RepLoop:
k+1
For j=1 To i-1 : If Y(j)=k : Goto RepLoop : EndIf : Next
If gcd(k,Y(i-2))=1 Or gcd(k,Y(i-1))>1 : Continue : EndIf
Y(i)=k : Break
ForEver
Next
For i=1 To 30 : Print(Str(Y(i))+" ") : Next : Input()
EndIf```
Output:
```1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17
```

## Python

Works with: Python version 3.7
```'''Yellowstone permutation OEIS A098550'''

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

from matplotlib import pyplot

# yellowstone :: [Int]
def yellowstone():
'''A non-finite stream of terms from
the Yellowstone permutation.
OEIS A098550.
'''
# relativelyPrime :: Int -> Int -> Bool
def relativelyPrime(a):
return lambda b: 1 == gcd(a, b)

# nextWindow :: (Int, Int, [Int]) -> (Int, Int, [Int])
def nextWindow(triple):
p2, p1, rest = triple
[rp2, rp1] = map(relativelyPrime, [p2, p1])

# match :: [Int] -> (Int, [Int])
def match(xxs):
x, xs = uncons(xxs)['Just']
return (x, xs) if rp1(x) and not rp2(x) else (
second(cons(x))(
match(xs)
)
)
n, residue = match(rest)
return (p1, n, residue)

return chain(
range(1, 3),
map(
itemgetter(1),
iterate(nextWindow)(
(2, 3, count(4))
)
)
)

# TEST ----------------------------------------------------
# main :: IO ()
def main():
'''Terms of the Yellowstone permutation.'''

print(showList(
take(30)(yellowstone())
))
pyplot.plot(
take(100)(yellowstone())
)
pyplot.xlabel(main.__doc__)
pyplot.show()

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

# Just :: a -> Maybe a
def Just(x):
'''Constructor for an inhabited Maybe (option type) value.
Wrapper containing the result of a computation.
'''
return {'type': 'Maybe', 'Nothing': False, 'Just': x}

# Nothing :: Maybe a
def Nothing():
'''Constructor for an empty Maybe (option type) value.
Empty wrapper returned where a computation is not possible.
'''
return {'type': 'Maybe', 'Nothing': True}

# cons :: a -> [a] -> [a]
def cons(x):
'''Construction of a list from x as head,
and xs as tail.
'''
return lambda xs: [x] + xs if (
isinstance(xs, list)
) else x + xs if (
isinstance(xs, str)
) else chain([x], xs)

# 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 go

# second :: (a -> b) -> ((c, a) -> (c, b))
def second(f):
'''A simple function lifted to a function over a tuple,
with f applied only to the second of two values.
'''
return lambda xy: (xy[0], f(xy[1]))

# showList :: [a] -> String
def showList(xs):
'''Stringification of a list.'''
return '[' + ','.join(repr(x) for x in xs) + ']'

# 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, tuple))
else list(islice(xs, n))
)

# uncons :: [a] -> Maybe (a, [a])
def uncons(xs):
'''The deconstruction of a non-empty list
(or generator stream) into two parts:
a head value, and the remaining values.
'''
if isinstance(xs, list):
return Just((xs[0], xs[1:])) if xs else Nothing()
else:
nxt = take(1)(xs)
return Just((nxt[0], xs)) if nxt else Nothing()

# MAIN ---
if __name__ == '__main__':
main()
```
Output:
`1,2,3,4,9,8,15,14,5,6,25,12,35,16,7,10,21,20,27,22,39,11,13,33,26,45,28,51,32,17]`

## Quackery

`gcd` is defined at Greatest common divisor#Quackery.

```  [ stack ]                 is seqbits      (   --> s )

[ bit
seqbits take |
seqbits put ]           is seqadd       ( n -->   )

[ bit
seqbits share & not ]   is notinseq     ( n --> b )

[ temp put
' [ 1 2 3 ]
7 seqbits put
4
[ dip
[ dup -1 peek
over -2 peek ]
dup dip
[ tuck gcd 1 !=
unrot gcd 1 =
and ]
swap if
[ dup dip join
3 ]
[ 1+
dup notinseq until ]
over size temp share
< not until ]
drop
seqbits release
temp take split drop ] is yellowstones ( n --> [ )

30 yellowstones echo```
Output:
`[ 1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17 ]`

## Racket

```#lang racket

(require plot)

(define a098550
(let ((hsh# (make-hash '((1 . 1) (2 . 2) (3 . 3))))
(rev# (make-hash '((1 . 1) (2 . 2) (3 . 3)))))
(λ (n)
(hash-ref hsh# n
(λ ()
(let ((a_n (for/first ((i (in-naturals 4))
#:unless (hash-has-key? rev# i)
#:when (and (= (gcd i (a098550 (- n 1))) 1)
(> (gcd i (a098550 (- n 2))) 1)))
i)))
(hash-set! hsh# n a_n)
(hash-set! rev# a_n n)
a_n))))))

(map a098550 (range 1 (add1 30)))

(plot (points
(map (λ (i) (vector i (a098550 i))) (range 1 (add1 100)))))
```
Output:

Just the output text... you'll have to run this yourself in racket to see the plot!

`'(1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17)`

## Raku

(formerly Perl 6)

Works with: Rakudo version 2020.01

Not really clear whether a line graph or bar graph was desired, so generate both. Also, 100 points don't really give a good feel for the overall shape so do 500.

```my @yellowstone = 1, 2, 3, -> \$q, \$p {
state @used = True xx 4;
state \$min  = 3;
my \index = (\$min .. *).first: { not @used[\$_] and \$_ gcd \$q != 1 and \$_ gcd \$p == 1 };
@used[index] = True;
\$min = @used.first(!*, :k) // +@used - 1;
index
} … *;

put "The first 30 terms in the Yellowstone sequence:\n", @yellowstone[^30];

use SVG;
use SVG::Plot;

my @x = ^500;

my \$chart = SVG::Plot.new(
background  => 'white',
width       => 1000,
height      => 600,
plot-width  => 950,
plot-height => 550,
x           => @x,
x-tick-step => { 10 },
y-tick-step => { 50 },
min-y-axis  => 0,
values      => [@yellowstone[@x],],
title       => "Yellowstone Sequence - First {+@x} values (zero indexed)",
);

my \$line = './Yellowstone-sequence-line-perl6.svg'.IO;
my \$bars = './Yellowstone-sequence-bars-perl6.svg'.IO;

\$line.spurt: SVG.serialize: \$chart.plot: :lines;
\$bars.spurt: SVG.serialize: \$chart.plot: :bars;
```
Output:
```The first 30 terms in the Yellowstone sequence:
1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17```

See (offsite SVG images) Line graph or Bar graph

## REXX

### horizontal list of numbers

```/*REXX program calculates any number of terms in the Yellowstone (permutation) sequence.*/
parse arg m .                                    /*obtain optional argument from the CL.*/
if m=='' | m==","  then m= 30                    /*Not specified?  Then use the default.*/
!.= 0                                            /*initialize an array of numbers(used).*/
# = 0                                            /*count of Yellowstone numbers in seq. */
\$=                                               /*list   "      "         "     "  "   */
do j=1  until #==m;  prev= # - 1
if j<5  then do;  #= #+1;   @.#= j;  !.#= j;  !.j= 1;  \$= strip(\$ j);  iterate;  end

do k=1;   if !.k  then iterate          /*Already used?  Then skip this number.*/
if gcd(k, @.prev)<2  then iterate       /*Not meet requirement?  Then skip it. */
if gcd(k, @.#) \==1  then iterate       /* "    "       "          "    "   "  */
#= #+1;   @.#= k;     !.k= 1;   \$= \$ k  /*bump ctr; assign; mark used; add list*/
leave                                   /*find the next Yellowstone seq. number*/
end   /*k*/
end      /*j*/
say \$                                            /*display a list of a Yellowstone seq. */
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
gcd: parse arg x,y;  do until y==0;  parse value  x//y  y   with   y  x;  end;    return x
```
output   when using the default input:     30
```1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17
```

### vertical histogram plot

A horizontal histogram could also be shown,   but it would require a taller (higher) plot with more vertical screen real estate.

```/*REXX program calculates any number of terms in the Yellowstone (permutation) sequence.*/
parse arg m .                                    /*obtain optional argument from the CL.*/
if m=='' | m==","  then m= 30                    /*Not specified?  Then use the default.*/
!.= 0                                            /*initialize an array of numbers(used).*/
# = 0                                            /*count of Yellowstone numbers in seq. */
\$ =                                              /*list   "      "         "     "  "   */
do j=1  until #==m;         prev= # - 1
if j<5  then do;  #= #+1;   @.#= j;  !.#= j;  !.j= 1;  \$= strip(\$ j);  iterate;  end

do k=1;   if !.k  then iterate          /*Already used?  Then skip this number.*/
if gcd(k, @.prev)<2  then iterate       /*Not meet requirement?  Then skip it. */
if gcd(k, @.#) \==1  then iterate       /* "    "       "          "    "   "  */
#= # + 1; @.#= k;     !.k= 1;   \$= \$ k  /*bump ctr; assign; mark used; add list*/
leave                                   /*find the next Yellowstone seq. number*/
end   /*k*/
end      /*j*/

call \$histo  \$   '(vertical)'                    /*invoke a REXX vertical histogram plot*/
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
gcd: parse arg x,y;  do until y==0;  parse value  x//y  y   with   y  x;  end;    return x
```
output   when using the input:     532

The plot is shown at three─quarter scale.

```                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            ■
│
│
│
│
│
│
│
│
│
│
■                                                                                                            │
│                                                                                                            │
│                                                                                                            │
■                    │                                                                                                            │
│                    │                                                                                                            │
│                    │                                                                                                            │
│                    │                                                                                                            │
■                                     │                    │                                                                           ■                                │
│                                     │                    │                                                                ■          │                                │
│                                     │                    │                                                                │          │                                │
■                              │                                     │                    │                                                                │          │                                │
│                              │                                     │                    │                                                                │          │                                │
│                              │                                     │                    │               ■                                                │          │                                │
■                      │                              │                                     │                    │               │                                                │          │                                │
│                      │                              │                                     │                    │               │                                                │          │                                │
│                      │                              │                                     │                    │               │                                                │          │                                │
│                      │                              │                              ■      │                    │               │                                                │          │                                │
│                      │                              │                              │      │                    │               │                                                │          │                                │
│                      │                              │        ■                     │      │                    │               │                                                │          │                                │
│                      │             ■                │        │                     │      │                    │               │                                                │          │                                │
│                      │    ■        │                │        │                     │      │                    │               │                                                │          │                                │
│                      │    │        │                │        │                     │      │                    │               │                                                │          │                                │    ■
│                      │    │        │                │        │                     │      │                    │               │                                                │          │               ■           ■    │    │
■                                                                                                │                      │    │        │                │        │                     │      │                    │               │                                 ■       ■      │          │ ■ ■  ■        │         ■ │    │ ■  │
│                                                                              ■                 │                      │    │        │                │        │                     │      │                    │               │                                 │     ■ │      │     ■    │ │ │  │ ■      │ ■   ■   │ │ ■  │ │  │ ■
│                                                                              │                 │                      │    │        │                │        │                     │      │               ■    │               │                     ■     ■     │     │ │ ■ ■  │ ■ ■ │    │ │ │  │ │ ■    │ │ ■ │ ■ │ │ │  │ │  │ │
■                            │                                                                              │                 │                      │    │        │                │        │                     │      │               │    │   ■    ■      │    ■           ■    │ ■ ■ │   ■ │ ■ ■ │ │ │ │  │ │ │ │    │ │ │  │ │ │    │ │ │ │ │ │ │ │  │ │  │ │
│                            │                                          ■                                   │                 │                      │    │        │                │        │                     │   ■  │         ■     │    │ ■ │ ■  │      │ ■  │         ■ │    │ │ │ │ ■ │ │ │ │ │ │ │ │  │ │ │ │ ■  │ │ │  │ │ │    │ │ │ │ │ │ │ │  │ │  │ │
│                            │                                 ■        │                                   │                 │                      │    │        │                │        │       ■             │ ■ │  │     ■   │   ■ │    │ │ │ │  │      │ │  │ ■ ■ ■ ■ │ │ ■  │ │ │ │ │ │ │ │ │ │ │ │ │  │ │ │ │ │  │ │ │  │ │ │ ■  │ │ │ │ │ │ │ │  │ │  │ │
│                            │          ■                      │        │                                   │                 │                      │    │        │             ■  │        │     ■ │  ■       ■  │ │ │  │ ■ ■ │ ■ │ ■ │ │    │ │ │ │  │ ■ ■  │ │  │ │ │ │ │ │ │ │  │ │ │ │ │ │ │ │ │ │ │ │ │  │ │ │ │ │  │ │ │  │ │ │ │ ■│ │ │ │ │ │ │ │  │ │  │ │
│                            │          │                      │        │                                   │                 │   ■                  │    │   ■    │ ■         ■ │  │ ■      │ ■ ■ │ │  │   ■ ■ │  │ │ │  │ │ │ │ │ │ │ │ │ ■  │ │ │ │  │ │ │  │ │  │ │ │ │ │ │ │ │  │ │ │ │ │ │ │ │ │ │ │ │ │  │ │ │ │ │ ■│ │ │  │ │■│■│ ││■│■│ │■│ │■│■│ ■│■│ ■│■│
│                            │          │                      │        │                 ■                 │                 │   │       ■       ■  │ ■  │ ■ │    │ │       ■ │ │  │ │ ■ ■  │ │ │ │ │  │ ■ │ │ │  │ │ │  │ │ │ │ │ │ │ │ │ │  │ │ │ │  │ │ │  │ │  │ │ │ │ │ │ │ │ ■│ │ │ │■│ │ │ │ │ │ │■│■│ ■│■│■│■│■│ ││■│■│ ■│■│││││ ││││││■│││■│││││ ││││ ││││
■                                          │                            │          │                      │        │                 │                 │     ■      ■    │ ■ │       │   ■ ■ │  │ │  │ │ │ ■  │ │ ■ ■ ■ │ │ │  │ │ │ │  │ │ │ │ │  │ │ │ │ │  │ │ │  │ │ │ │ │ │ │ │ │ │  │ │ │ │  │ │ │  │ │  │ │■│■│■│■│■│■│ ││■│■│■│││ │■│■│■│■│■│││││ ││││││││││ ││││││ ││││││││ ││││││││││││││││ ││││ ││││
│                                          │                            │          │                      │        │             ■ ■ │        ■   ■    │   ■ │ ■    │ ■  │ │ │ ■     │ ■ │ │ │  │ │  │ │ │ │  │ │ │ │ │ │ │ │  │ │ │ │  │ │ │ │ │  │ │ │ │ │  │ │ │  │ │ │ │ │ │■│■│■│■│ ■│ │■│■│ ■│■│■│ ■│■│ ■│■│││││││││││││ ││││││││││■│││││││││││││││ ││││││││││ ││││││ ││││││││ ││││││││││││││││ ││││ ││││
│                         ■                │                            │          │                      │        │     ■ ■     │ │ │   ■    │ ■ │ ■  │ ■ │ │ │ ■  │ │  │ │ │ │   ■ │ │ │ │ │  │ │  │ │ │ │  │ │ │ │ │ │ │ │  │ │ │ │  │ │ │ │ │  │■│■│■│■│ ■│■│■│ ■│■│■│■│■│■│││││││││ ││■│││││ ││││││ ││││ ││││││││││││││││ ││││││││││││││││││││││││││ ││││││││││ ││││││ ││││││││ ││││││││││││││││ ││││ ││││
■                              │         ■               │                │                            │          │              ■  ■    │   ■ ■  │ ■ ■ │ │ ■ ■ │ │ │ ■ │    │ │ │ │  │ │ │ │ │ │  │ │  │ │ │ │ ■ │ │ │ │ │ │  │ │  │ │ │ │  │ │ │■│■│ │■│■│ ■│■│■│■│ ■│■│■│■│■│ ■│││││││││ ││││││ ││││││││││││││││││││ ││││││││ ││││││ ││││ ││││││││││││││││ ││││││││││││││││││││││││││ ││││││││││ ││││││ ││││││││ ││││││││││││││││ ││││ ││││
│                              │         │               │                │                        ■   │   ■ ■    │ ■    ■ ■   ■ │  │ ■  │ ■ │ │  │ │ │ │ │ │ │ │ │ │ │ │    │ │ │ │  │ │ │ │ │ │  │ │  │ │ │ │ │ │■│■│■│■│■│ ■│■│ ■│■│■│■│ ■│■│■│││││■│││││ ││││││││ ││││││││││ ││││││││││ ││││││ ││││││││││││││││││││ ││││││││ ││││││ ││││ ││││││││││││││││ ││││││││││││││││││││││││││ ││││││││││ ││││││ ││││││││ ││││││││││││││││ ││││ ││││
│            ■                 │         │               │                │             ■    ■ ■ ■ │   │ ■ │ │ ■  │ │ ■  │ │   │ │  │ │  │ │ │ │  │ │ │ │ │ │ │ │ │ │ │ │ ■ ■│ │ │■│  │ │■│■│■│■│ ■│■│ ■│■│■│■│■│■│││││││││││ ││││ ││││││││ ││││││││││││││││ ││││││││ ││││││││││ ││││││││││ ││││││ ││││││││││││││││││││ ││││││││ ││││││ ││││ ││││││││││││││││ ││││││││││││││││││││││││││ ││││││││││ ││││││ ││││││││ ││││││││││││││││ ││││ ││││
│            │                 │         │     ■         │        ■ ■ ■   │   ■ ■     ■ │ ■  │ │ │ │   │ │ │ │ │  │ │ │  │ │ ■ │ │  │ │  │ │ │ │  │■│ │■│■│■│■│■│■│■│■│■│■│ ││■│■│││ ■│■│││││││││ ││││ ││││││││││││││││││││││ ││││ ││││││││ ││││││││││││││││ ││││││││ ││││││││││ ││││││││││ ││││││ ││││││││││││││││││││ ││││││││ ││││││ ││││ ││││││││││││││││ ││││││││││││││││││││││││││ ││││││││││ ││││││ ││││││││■││││││││││││││││■││││■││││
│ ■          │                 │         │ ■ ■ │ ■  ■ ■  │ ■ ■ ■  │ │ │   │ ■ │ │ ■   │ │ │  │ │ │ │   │ │ │ │ │  │■│■│  │■│■│■│■│ ■│■│ ■│■│■│■│ ■│││■│││││││││││││││││││││ ││││││││ ││││││││││││ ││││ ││││││││││││││││││││││ ││││ ││││││││ ││││││││││││││││ ││││││││ ││││││││││ ││││││││││ ││││││ ││││││││││││││││││││ ││││││││ ││││││ ││││ ││││││││││││││││■││││││││││││││││││││││││││■││││││││││■││││││■│││││││││││││││││││││││││││││││││││
■                          │ │          │ ■ ■     ■  ■    │ ■ ■ ■   │ │ │ │ │  │ │  │ │ │ │  │ │ │   │ │ │ │ │   │■│■│ ■│■│■│■│■ ■│■│■│■│■│ ■│││││ ■│││││││││ ││││ ││││││││ ││││││││││││││││││││││││││ ││││││││ ││││││││││││ ││││ ││││││││││││││││││││││ ││││ ││││││││ ││││││││││││││││ ││││││││ ││││││││││ ││││││││││■││││││■││││││││││││││││││││■││││││││■││││││■││││■│││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││
■ │                     ■ ■  │ │ ■   ■ ■  │ │ │     │  │ ■  │ │ │ │   │ │ │ │ │  │■│  │■│■│■│ ■│■│■│■ ■│■│■│■│■│■■■│││││ │││││││││ ││││││││││ ││││││ ││││││││││ ││││ ││││││││ ││││││││││││││││││││││││││ ││││││││ ││││││││││││ ││││ ││││││││││││││││││││││■││││■││││││││■││││││││││││││││■││││││││■││││││││││■│││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││
■ ■ │ │      ■ ■    ■ ■ ■ ■ │ │  │ │ │ ■ │ │  │ │ │ ■ ■■│  │■│ ■│■│■│■│■ ■│■│■│■│■│ ■│││ ■│││││││ │││││││ ││││││││││││││││││ │││││││││ ││││││││││ ││││││ ││││││││││ ││││ ││││││││ ││││││││││││││││││││││││││■││││││││■││││││││││││■││││■│││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││
■      ■       ■          ■ ■   │ │ │ │ ■  ■ │ │    │■│■│ │ │■│ ■│■│■│■│■│■│ ■│■│■│■│■│││ ■│││ │││││││││ ││││││││││ ││││ ││││││││ │││││││ ││││││││││││││││││■│││││││││■││││││││││■││││││■││││││││││■││││■││││││││■│││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││
■   ■ │   ■  │   ■   │ ■ ■ ■■   │■│■■ │■│■│■│■│ ■│■│■│■  ■│││││■│■│││ ││││││││││││ ││││││││││││■││││■│││││││││■││││││││││■││││■││││││││■│││││││■│││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││
────────────┴───────┴──┴─┴─┴┴──┴─┴┴┴┴┴┴┴┴┴─┴┴┴┴┴┴┴─┴┴┴┴┴┴┴┴┴─┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴
1234981156213171222231132425311825363934361494546524157485862359696751616171713717181844818191718191493921111911151211111111115111161411111216131212121271212713121212712121218151212121211121212812121212191612131322115222212322222223123231252323231252323232323232423242323221232323231262323242323124231292424232323242324242413134137343434137343434343434341313434341383434353513534343435138353513134353535353435363413135353513535351393514545454545454545246454646454546464546464646241464646452414646246464645256464646464647462414624746
54  5256 0107291336581278545456109839254709038125616729183498741036627407016102613228322844546702600404850730608191412189172512228283037233031334214140346415350953646473585164657266747775986868799978788909107090090800009151912022100142024221263333353641344047454455582785668673555569606877381793777186898286939992919340004505040611518171222201527296312834356333748414274155575850749536586269617371667972828899375898699780919989790903031414171823143292030350806102824373241231424354552615857248636262309757253716871763868948285
5                9 3 1 5 5  1 7 3    9 5 5 5 5 5  9 30741 361 258525634187 0729 074389016 6525458525 0367 45892189 4703490 016725652189476389 274189016 0967214525185456530987230945701839652545673852903618349410327658307497216907251258545658541701839036329496381072145658590421165458547725610309278143653048327437658545256530909276981985452510667810927416387092110343456732987437658590741670329638523123418309612985456309214765381458525497836585907094169369012307412965839072314387
3                              1                                     7                    7                                                                5          5                                3
```

## Ring

```see "working..." + nl
row = 3
num = 2
numbers = 1:51
first = 2
second = 3
see "Yellowstone numbers are:" + nl
see "1 " + first + " " + second + " "

for n = 4 to len(numbers)
flag1 = 1
flag2 = 1
if first < numbers[n]
min = first
else
min = numbers[n]
ok
for m = 2 to min
if first%m = 0 and numbers[n]%m = 0
flag1 = 0
exit
ok
next
if second < numbers[n]
min = second
else
min = numbers[n]
ok
for m = 2 to min
if second%m = 0 and numbers[n]%m = 0
flag2 = 0
exit
ok
next
if flag1 = 0 and flag2 = 1
see "" + numbers[n] + " "
first = second
second = numbers[n]
del(numbers,n)
row = row+1
if row%10 = 0
see nl
ok
num = num + 1
if num = 29
exit
ok
n = 3
ok
next

see "Found " + row + " Yellowstone numbers" + nl
see "done..." + nl```
Output:
```working...
Yellowstone numbers are:
1 2 3 4 9 8 15 14 5 6
25 12 35 16 7 10 21 20 27 22
39 11 13 33 26 45 28 51 32 17
Found 30 Yellowstone numbers
done...
```

## RPL

Works with: Halcyon Calc version 4.2.7
```≪
IF DUP2 < THEN SWAP END
WHILE DUP REPEAT SWAP OVER MOD END
DROP
≫ 'GCD' STO

≪
DUP SIZE 1 - GETI ROT ROT GET → am2 am1
≪ 3 DO
DO 1 + UNTIL DUP2 POS NOT END
UNTIL DUP am1 GCD 1 == OVER am2 GCD 1 ≠ AND END
+
≫ ≫ 'YELLO' STO
```
```( a b -- gcd(a,b) )
Ensure a > b
Euclidean algorithm

( { a(1)..a(n-1) } -- { a(1)..a(n) } )
Store locally a(n-1) and a(n-2)

Find smallest number not already in sequence
Check primality requirements

```

The following words in the command line deliver what is required:

```{ 1 2 3 }
≪ 1 27 START YELLO NEXT ≫ EVAL
```
Output:
```1: { 1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17 ]
```

## Ruby

```def yellow(n)
a = [1, 2, 3]
b = { 1 => true, 2 => true, 3 => true }
i = 4
while n > a.length
if !b[i] && i.gcd(a[-1]) == 1 && i.gcd(a[-2]) > 1
a << i
b[i] = true
i = 4
end
i += 1
end
a
end

p yellow(30)
```
Output:
```[1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17]
```

## Rust

```// [dependencies]
// num = "0.3"
// plotters = "^0.2.15"

use num::integer::gcd;
use plotters::prelude::*;
use std::collections::HashSet;

fn yellowstone_sequence() -> impl std::iter::Iterator<Item = u32> {
let mut sequence: HashSet<u32> = HashSet::new();
let mut min = 1;
let mut n = 0;
let mut n1 = 0;
let mut n2 = 0;
std::iter::from_fn(move || {
n2 = n1;
n1 = n;
if n < 3 {
n += 1;
} else {
n = min;
while !(!sequence.contains(&n) && gcd(n1, n) == 1 && gcd(n2, n) > 1) {
n += 1;
}
}
sequence.insert(n);
while sequence.contains(&min) {
sequence.remove(&min);
min += 1;
}
Some(n)
})
}

// Based on the example in the "Quick Start" section of the README file for
// the plotters library.
fn plot_yellowstone(filename: &str) -> Result<(), Box<dyn std::error::Error>> {
let root = BitMapBackend::new(filename, (800, 600)).into_drawing_area();
root.fill(&WHITE)?;
let mut chart = ChartBuilder::on(&root)
.caption("Yellowstone Sequence", ("sans-serif", 24).into_font())
.margin(10)
.x_label_area_size(20)
.y_label_area_size(20)
.build_ranged(0usize..100usize, 0u32..180u32)?;
chart.configure_mesh().draw()?;
chart.draw_series(LineSeries::new(
yellowstone_sequence().take(100).enumerate(),
&BLUE,
))?;
Ok(())
}

fn main() {
println!("First 30 Yellowstone numbers:");
for y in yellowstone_sequence().take(30) {
print!("{} ", y);
}
println!();
match plot_yellowstone("yellowstone.png") {
Ok(()) => {}
Err(error) => eprintln!("Error: {}", error),
}
}
```
Output:

A plot of the first 100 Yellowstone numbers is saved to the file "yellowstone.png".

```First 30 Yellowstone numbers:
1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17
```

## Scala

Translation of: Java
```import scala.util.control.Breaks._

object YellowstoneSequence extends App {

println(s"First 30 values in the yellowstone sequence:\n\${yellowstoneSequence(30)}")

def yellowstoneSequence(sequenceCount: Int): List[Int] = {
var yellowstoneList = List(1, 2, 3)
var num = 4
var notYellowstoneList = List[Int]()

while (yellowstoneList.size < sequenceCount) {
val foundIndex = notYellowstoneList.indexWhere(test =>
gcd(yellowstoneList(yellowstoneList.size - 2), test) > 1 &&
gcd(yellowstoneList.last, test) == 1
)

if (foundIndex >= 0) {
yellowstoneList = yellowstoneList :+ notYellowstoneList(foundIndex)
notYellowstoneList = notYellowstoneList.patch(foundIndex, Nil, 1)
} else {
breakable({
while (true) {
if (gcd(yellowstoneList(yellowstoneList.size - 2), num) > 1 &&
gcd(yellowstoneList.last, num) == 1) {
yellowstoneList = yellowstoneList :+ num
num += 1
// break the inner while loop
break
}
notYellowstoneList = notYellowstoneList :+ num
num += 1
}
});
}
}
yellowstoneList
}

def gcd(a: Int, b: Int): Int = if (b == 0) a else gcd(b, a % b)
}
```
Output:
```First 30 values in the yellowstone sequence:
List(1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17)

```

## Tcl

```proc gcd {a b} {
while {\$b} {
lassign [list \$b [expr {\$a % \$b}]] a b
}
return \$a
}

proc gen_yellowstones {{maxN 30}} {
set r {}
for {set n 1} {\$n <= \$maxN} {incr n} {
if {\$n <= 3} {
lappend r \$n
} else {
## NB: list indices start at 0, not 1.
set pred    [lindex \$r end  ]       ;# a(n-1): coprime
set prepred [lindex \$r end-1]       ;# a(n-2): not coprime
for {set k 4} {1} {incr k} {
if {[lsearch -exact \$r \$k] >= 0} { continue }
if {1 != [gcd \$k \$pred   ]} { continue }
if {1 == [gcd \$k \$prepred]} { continue }
## candidate k survived all tests...
break
}
lappend r \$k
}
}
return \$r
}
puts "The first 30 Yellowstone numbers are:"
puts [gen_yellowstones]
```
Output:
```The first 30 Yellowstone numbers are:
1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17
```

## uBasic/4tH

Translation of: VBA
Works with: v3.64.0
```Dim @y(30)

@y(0) = 1
@y(1) = 2
@y(2) = 3

For i = 3 To 29
k = 3
Do
k = k + 1
If (FUNC(_gcd(k, @y(i-2))) = 1) + (FUNC(_gcd(k, @y(i-1))) > 1) Then
Continue
EndIf

For j = 0 To i - 1
If @y(j) = k Then Unloop : Continue
Next

@y(i) = k : Break
Loop
Next

For i = 0 To 29
Print @y(i); " ";
Next

Print : End

_gcd Param (2)
If b@ = 0 Then Return (a@)
Return (FUNC(_gcd(b@, a@ % b@)))
```
Output:
```1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17

0 OK, 0:670
```

## Uiua

Very simple approach, probably lots of room for improvement. Run it in Uiua Pad to see the plot.

```Gcd ← ⊙◌⍢(⟜◿:|±,)
NextY ← ⍢(+1|⟨¬≍0_1≡(=1Gcd)⊙(↙¯2)|1⟩∊,,)1 # [a0 a1...an] -> an+1
N ← 200
⍢(⊂:NextY|<N⧻)[1 2 3]
⟜(↙30)      # First 30
▽⟜≡▽2⊞= ⇌⇡N # Plot 200```
Output:
```[1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17]
```

## VBA

```Function gcd(a As Long, b As Long) As Long
If b = 0 Then
gcd = a
Exit Function
End If
gcd = gcd(b, a Mod b)
End Function

Sub Yellowstone()
Dim i As Long, j As Long, k As Long, Y(1 To 30) As Long

Y(1) = 1
Y(2) = 2
Y(3) = 3

For i = 4 To 30
k = 3
Do
k = k + 1
If gcd(k, Y(i - 2)) = 1 Or gcd(k, Y(i - 1)) > 1 Then GoTo EndLoop:
For j = 1 To i - 1
If Y(j) = k Then GoTo EndLoop:
Next j
Y(i) = k
Exit Do
EndLoop:
Loop
Next i

For i = 1 To 30
Debug.Print Y(i) & " ";
Next i
End Sub
```
Output:
```1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17
```

## V (Vlang)

Translation of: go
```fn gcd(xx int, yy int) int {
mut x := xx
mut y := yy
for y != 0 {
x, y = y, x%y
}
return x
}

fn yellowstone(n int) []int {
mut m := map[int]bool{}
mut a := []int{len: n+1}
for i in 1..4 {
a[i] = i
m[i] = true
}
mut min := 4
for c := 4; c <= n; c++ {
for i := min; ; i++ {
if !m[i] && gcd(a[c-1], i) == 1 && gcd(a[c-2], i) > 1 {
a[c] = i
m[i] = true
if i == min {
min++
}
break
}
}
}
return a[1..]
}

fn main() {
mut x := []int{len: 100}
for i in 0..100 {
x[i] = i + 1
}
y := yellowstone(100)
println("The first 30 Yellowstone numbers are:")
println(y[..30])
}```
Output:
```The first 30 Yellowstone numbers are:
[1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17]
```

## Wren

Translation of: Go
Library: Wren-math

Without the extra credit part.

```import "./math" for Int

var yellowstone = Fn.new { |n|
var m = {}
var a = List.filled(n + 1, 0)
for (i in 1..3) {
a[i] = i
m[i] = true
}
var min = 4
for (c in 4..n) {
var i = min
while (true) {
if (!m[i] && Int.gcd(a[c-1], i) == 1 && Int.gcd(a[c-2], i) > 1) {
a[c] = i
m[i] = true
if (i == min) min = min + 1
break
}
i = i + 1
}
}
return a[1..-1]
}

var x = List.filled(30, 0)
for (i in 0...30) x[i] = i + 1
var y = yellowstone.call(30)
System.print("The first 30 Yellowstone numbers are:")
System.print(y)
```
Output:
```The first 30 Yellowstone numbers are:
[1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17]
```

## XPL0

```func GCD(N, D);         \Return the greatest common divisor of N and D
int  N, D, R;           \numerator, denominator, remainder
[if D > N then
[R:=D; D:=N; N:=R]; \swap D and N
while D > 0 do
[R:= rem(N/D);
N:= D;
D:= R;
];
return N;
];

int I, A(30+1), N, T;
[for I:= 1 to 3 do A(I):= I;                    \givens
N:= 4;
repeat  T:= 4;
loop    [if GCD(T, A(N-1)) = 1 and      \relatively prime
GCD(T, A(N-2)) # 1 then     \not relatively prime
[loop   [for I:= 1 to N-1 do \test if in sequence
if T = A(I) then quit;
quit;
];
if I = N then           \T is not in sequence so
[A(N):= T;          \ add it in
N:= N+1;
quit;
];
];
T:= T+1;                        \next trial
];
until   N > 30;
for N:= 1 to 30 do
[IntOut(0, A(N));  ChOut(0, ^ )];
\\for N:= 1 to 100 do Point(N, A(N));           \plot demonstration
]```
Output:
```1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17
```

## zkl

Translation of: Julia

This sequence is limited to the max size of a Dictionary, 64k

```fcn yellowstoneW{	// --> iterator
Walker.zero().tweak(fcn(a,b){
foreach i in ([1..]){
if(not b.holds(i) and i.gcd(a[-1])==1 and i.gcd(a[-2]) >1){
a.del(0).append(i);	// only keep last two terms
b[i]=True;
return(i);
}
}
}.fp(List(2,3), Dictionary(1,True, 2,True, 3,True))).push(1,2,3);
}```
```println("The first 30 entries of the Yellowstone permutation:");
yellowstoneW().walk(30).concat(", ").println();```
Output:
```The first 30 entries of the Yellowstone permutation:
1, 2, 3, 4, 9, 8, 15, 14, 5, 6, 25, 12, 35, 16, 7, 10, 21, 20, 27, 22, 39, 11, 13, 33, 26, 45, 28, 51, 32, 17
```

Plot using Gnuplot

```gnuplot:=System.popen("gnuplot","w");
gnuplot.writeln("unset key; plot '-'");
yellowstoneW().pump(1_000, gnuplot.writeln.fp(" "));  // " 1\n", " 2\n", ...
gnuplot.writeln("e");
gnuplot.flush();