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Ingy döt Net 2023-07-01 11:58:00 -04:00
parent 7387c8f97b
commit cb5bb5e222
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4
Task/Yellowstone-sequence/00-META.yaml Normal file
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---
category:
- Prime Numbers
from: http://rosettacode.org/wiki/Yellowstone_sequence

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Task/Yellowstone-sequence/00-TASK.txt Normal file
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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).
;Task
: Find and show as output the first &nbsp;'''30'''&nbsp; Yellowstone numbers.
;Extra
: Demonstrate how to plot, with x = n and y coordinate a(n), the first 100 Yellowstone numbers.
;Related tasks:
:* &nbsp; [https://rosettacode.org/wiki/Greatest_common_divisor Greatest common divisor].
:* &nbsp; [https://rosettacode.org/wiki/Plot_coordinate_pairs Plot coordinate pairs].
;See also:
:* &nbsp; The OEIS entry: &nbsp; [https://oeis.org/A098550 A098550 The Yellowstone permutation].
:* &nbsp; Applegate et al, 2015: The Yellowstone Permutation [https://arxiv.org/abs/1501.01669].
<br><br>

30
Task/Yellowstone-sequence/11l/yellowstone-sequence-1.11l Normal file
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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_++
.sequence_.add(.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()

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Task/Yellowstone-sequence/11l/yellowstone-sequence-2.11l Normal file
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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)
b.add(i)
i = 4
i++
R a
print(yellow(30))

60
Task/Yellowstone-sequence/ALGOL-68/yellowstone-sequence.alg Normal file
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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;
FOR j WHILE NOT found DO
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

55
Task/Yellowstone-sequence/Action-/yellowstone-sequence.action Normal file
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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

68
Task/Yellowstone-sequence/Ada/yellowstone-sequence.ada Normal file
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with Ada.Text_IO;
with Ada.Containers.Ordered_Sets;
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
new Ada.Containers.Ordered_Sets (Integer);
-- 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
use Ada.Text_IO;
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;

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Task/Yellowstone-sequence/Arturo/yellowstone-sequence.arturo Normal file
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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

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Task/Yellowstone-sequence/AutoHotkey/yellowstone-sequence.ahk Normal file
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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
}

44
Task/Yellowstone-sequence/C++/yellowstone-sequence.cpp Normal file
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#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;
}

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Task/Yellowstone-sequence/C/yellowstone-sequence.c Normal file
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#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;
}

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Task/Yellowstone-sequence/D/yellowstone-sequence.d Normal file
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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();
}

81
Task/Yellowstone-sequence/Delphi/yellowstone-sequence.delphi Normal file
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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');
Readln;
plot.Kill;
plot.Free;
end.

34
Task/Yellowstone-sequence/Factor/yellowstone-sequence.factor Normal file
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USING: accessors assocs colors.constants
combinators.short-circuit io kernel math prettyprint sequences
sets ui ui.gadgets ui.gadgets.charts ui.gadgets.charts.lines ;
: 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
add-gadget "Yellowstone numbers" open-window

17
Task/Yellowstone-sequence/Forth/yellowstone-sequence.fth Normal file
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: 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

35
Task/Yellowstone-sequence/FreeBASIC/yellowstone-sequence.basic Normal file
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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

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Task/Yellowstone-sequence/Go/yellowstone-sequence.go Normal file
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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()
}

20
Task/Yellowstone-sequence/Haskell/yellowstone-sequence-1.hs Normal file
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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

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Task/Yellowstone-sequence/Haskell/yellowstone-sequence-2.hs Normal file
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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
import qualified Graphics.SVGFonts.ReadFont as F
----------------- 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
sansR <- F.loadFont "SourceSansPro_R.svg"
sansRB <- F.loadFont "SourceSansPro_RB.svg"
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

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Until=: 2 :'u^:(0-:v)^:_'
assert 44 -: >:Until(>&43) 32 NB. increment until exceeding 43
gcd=: +.
coprime=: 1 = gcd
prepare=:1 2 3"_ NB. start with the vector 1 2 3
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

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Task/Yellowstone-sequence/J/yellowstone-sequence-2.j Normal file
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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.
}}

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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>();
yellowstoneList.add(1);
yellowstoneList.add(2);
yellowstoneList.add(3);
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 ) {
yellowstoneList.add(notYellowstoneList.remove(found));
yellowSize++;
}
else {
while ( true ) {
if ( gcd(yellowstoneList.get(yellowSize-2), num) > 1 && gcd(yellowstoneList.get(yellowSize-1), num) == 1 ) {
yellowstoneList.add(num);
yellowSize++;
num++;
break;
}
notYellowstoneList.add(num);
num++;
}
}
}
return yellowstoneList;
}
private static final int gcd(int a, int b) {
if ( b == 0 ) {
return a;
}
return gcd(b, a%b);
}
}

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(() => {
'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();
})();

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# 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 ));

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"The first 30 entries of the Yellowstone permutation:",
[limit(30;yellowstone)]

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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 haskey(b, i)
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)

49
Task/Yellowstone-sequence/Kotlin/yellowstone-sequence.kotlin Normal file
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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) {
yellowstoneList.add(notYellowstoneList.removeAt(found))
yellowSize++
} else {
while (true) {
if (gcd(yellowstoneList[yellowSize - 2], num) > 1 && gcd(
yellowstoneList[yellowSize - 1], num
) == 1
) {
yellowstoneList.add(num)
yellowSize++
num++
break
}
notYellowstoneList.add(num)
num++
}
}
}
return yellowstoneList
}
private fun gcd(a: Int, b: Int): Int {
return if (b == 0) {
a
} else gcd(b, a % b)
}

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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()

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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[%]

18
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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:
result.add candidate
present.incl candidate
while start in present: inc start
break
inc candidate
echo yellowstone(30)

24
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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()

49
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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;

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(phixonline)-->
<span style="color: #000080;font-style:italic;">--
-- demo\rosetta\Yellowstone_sequence.exw
--</span>
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #7060A8;">requires</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"1.0.2"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">yellowstone</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">N</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">},</span>
<span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #004600;">true</span><span style="color: #0000FF;">,</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">i</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">4</span>
<span style="color: #008080;">while</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;"><</span> <span style="color: #000000;">N</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">></span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">or</span> <span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]=</span><span style="color: #004600;">false</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">and</span> <span style="color: #7060A8;">gcd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[$])=</span><span style="color: #000000;">1</span>
<span style="color: #008080;">and</span> <span style="color: #7060A8;">gcd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[$-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])></span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">i</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">></span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">b</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #004600;">false</span><span style="color: #0000FF;">,</span><span style="color: #000000;">i</span><span style="color: #0000FF;">-</span><span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">b</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">true</span>
<span style="color: #000000;">i</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">4</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">i</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"The first 30 entries of the Yellowstone permutation:\n%v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">yellowstone</span><span style="color: #0000FF;">(</span><span style="color: #000000;">30</span><span style="color: #0000FF;">)})</span>
<span style="color: #000080;font-style:italic;">-- a simple plot:</span>
<span style="color: #008080;">include</span> <span style="color: #000000;">pGUI</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
<span style="color: #008080;">include</span> <span style="color: #7060A8;">IupGraph</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">get_data</span><span style="color: #0000FF;">(</span><span style="color: #004080;">Ihandle</span> <span style="color: #000000;">graph</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">y500</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">yellowstone</span><span style="color: #0000FF;">(</span><span style="color: #000000;">500</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">w</span><span style="color: #0000FF;">,</span><span style="color: #000000;">h</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">IupGetIntInt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">graph</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"DRAWSIZE"</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">IupSetInt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">graph</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"XTICK"</span><span style="color: #0000FF;">,</span><span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">w</span><span style="color: #0000FF;"><</span><span style="color: #000000;">640</span><span style="color: #0000FF;">?</span><span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">h</span><span style="color: #0000FF;"><</span><span style="color: #000000;">300</span><span style="color: #0000FF;">?</span><span style="color: #000000;">100</span><span style="color: #0000FF;">:</span><span style="color: #000000;">50</span><span style="color: #0000FF;">):</span><span style="color: #000000;">20</span><span style="color: #0000FF;">))</span>
<span style="color: #7060A8;">IupSetInt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">graph</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"YTICK"</span><span style="color: #0000FF;">,</span><span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">h</span><span style="color: #0000FF;"><</span><span style="color: #000000;">250</span><span style="color: #0000FF;">?</span><span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">h</span><span style="color: #0000FF;"><</span><span style="color: #000000;">140</span><span style="color: #0000FF;">?</span><span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">h</span><span style="color: #0000FF;"><</span><span style="color: #000000;">120</span><span style="color: #0000FF;">?</span><span style="color: #000000;">700</span><span style="color: #0000FF;">:</span><span style="color: #000000;">350</span><span style="color: #0000FF;">):</span><span style="color: #000000;">200</span><span style="color: #0000FF;">):</span><span style="color: #000000;">100</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{{</span><span style="color: #7060A8;">tagset</span><span style="color: #0000FF;">(</span><span style="color: #000000;">500</span><span style="color: #0000FF;">),</span><span style="color: #000000;">y500</span><span style="color: #0000FF;">,</span><span style="color: #004600;">CD_RED</span><span style="color: #0000FF;">}}</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #7060A8;">IupOpen</span><span style="color: #0000FF;">()</span>
<span style="color: #004080;">Ihandle</span> <span style="color: #000000;">graph</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">IupGraph</span><span style="color: #0000FF;">(</span><span style="color: #000000;">get_data</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"RASTERSIZE=960x600"</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">IupSetAttributes</span><span style="color: #0000FF;">(</span><span style="color: #000000;">graph</span><span style="color: #0000FF;">,</span><span style="color: #008000;">`GTITLE="Yellowstone Numbers"`</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">IupSetInt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">graph</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"TITLESTYLE"</span><span style="color: #0000FF;">,</span><span style="color: #004600;">CD_ITALIC</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">IupSetAttributes</span><span style="color: #0000FF;">(</span><span style="color: #000000;">graph</span><span style="color: #0000FF;">,</span><span style="color: #008000;">`XNAME="n", YNAME="a(n)"`</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">IupSetAttributes</span><span style="color: #0000FF;">(</span><span style="color: #000000;">graph</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"XTICK=20,XMIN=0,XMAX=500"</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">IupSetAttributes</span><span style="color: #0000FF;">(</span><span style="color: #000000;">graph</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"YTICK=100,YMIN=0,YMAX=1400"</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">Ihandle</span> <span style="color: #000000;">dlg</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">IupDialog</span><span style="color: #0000FF;">(</span><span style="color: #000000;">graph</span><span style="color: #0000FF;">,</span><span style="color: #008000;">`TITLE="Yellowstone Names"`</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">IupSetAttributes</span><span style="color: #0000FF;">(</span><span style="color: #000000;">dlg</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"MINSIZE=290x140"</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">IupShow</span><span style="color: #0000FF;">(</span><span style="color: #000000;">dlg</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">platform</span><span style="color: #0000FF;">()!=</span><span style="color: #004600;">JS</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">IupMainLoop</span><span style="color: #0000FF;">()</span>
<span style="color: #7060A8;">IupClose</span><span style="color: #0000FF;">()</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<!--

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Task/Yellowstone-sequence/Phixmonti/yellowstone-sequence.phixmonti Normal file
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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 ?

16
Task/Yellowstone-sequence/PicoLisp/yellowstone-sequence.l Normal file
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(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))

18
Task/Yellowstone-sequence/PureBasic/yellowstone-sequence.basic Normal file
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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

147
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'''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()

33
Task/Yellowstone-sequence/Quackery/yellowstone-sequence.quackery Normal file
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[ 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
seqadd
3 ]
[ 1+
dup notinseq until ]
over size temp share
< not until ]
drop
seqbits release
temp take split drop ] is yellowstones ( n --> [ )
30 yellowstones echo

20
Task/Yellowstone-sequence/REXX/yellowstone-sequence-1.rexx Normal file
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/*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

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/*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

23
Task/Yellowstone-sequence/Racket/yellowstone-sequence.rkt Normal file
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#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)))))

35
Task/Yellowstone-sequence/Raku/yellowstone-sequence.raku Normal file
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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;

53
Task/Yellowstone-sequence/Ring/yellowstone-sequence.ring Normal file
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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

16
Task/Yellowstone-sequence/Ruby/yellowstone-sequence.rb Normal file
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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)

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// [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),
}
}

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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]

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Task/Yellowstone-sequence/V-(Vlang)/yellowstone-sequence.v Normal file
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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])
}

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Task/Yellowstone-sequence/VBA/yellowstone-sequence.vba Normal file
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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

30
Task/Yellowstone-sequence/Wren/yellowstone-sequence.wren Normal file
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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)

35
Task/Yellowstone-sequence/XPL0/yellowstone-sequence.xpl0 Normal file
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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
]

11
Task/Yellowstone-sequence/Zkl/yellowstone-sequence-1.zkl Normal file
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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);
}

2
Task/Yellowstone-sequence/Zkl/yellowstone-sequence-2.zkl Normal file
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println("The first 30 entries of the Yellowstone permutation:");
yellowstoneW().walk(30).concat(", ").println();

6
Task/Yellowstone-sequence/Zkl/yellowstone-sequence-3.zkl Normal file
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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();
ask("Hit return to finish"); gnuplot.close();