Yellowstone sequence: Difference between revisions
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=={{header|PureBasic}}== |
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<lang PureBasic>Procedure.i gcd(x.i,y.i) |
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While y<>0 : t=x : x=y : y=t%y : Wend : ProcedureReturn x |
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EndProcedure |
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If OpenConsole() |
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Dim Y.i(100) |
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For i=1 To 100 |
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If i<=3 : Y(i)=i : Continue : EndIf : k=3 |
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Repeat |
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RepLoop: |
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k+1 |
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For j=1 To i-1 : If Y(j)=k : Goto RepLoop : EndIf : Next |
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If gcd(k,Y(i-2))=1 Or gcd(k,Y(i-1))>1 : Continue : EndIf |
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Y(i)=k : Break |
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ForEver |
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Next |
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For i=1 To 30 : Print(Str(Y(i))+" ") : Next : Input() |
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EndIf</lang> |
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{{out}} |
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<pre>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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</pre> |
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Revision as of 12:09, 6 May 2021
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).
- Task
- 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.
- Related tasks
- See also
-
- The OEIS entry: A098550 The Yellowstone permutation.
- Applegate et al, 2015: The Yellowstone Permutation [1].
C
<lang c>#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;
}</lang>
- 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++
<lang cpp>#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;
}</lang>
- 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
<lang d>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();
}</lang>
- 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
Boost.Generics.Collection and Boost.Process are part of DelphiBoostLib. <lang Delphi> 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.</lang>
- 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
Factor
<lang factor>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</lang>
- 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
FreeBASIC
<lang 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</lang>
- 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. <lang go>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()
}</lang>
- 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]
Haskell
<lang haskell>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</lang>
- 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:
<lang haskell>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</lang>
- 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
<lang J> 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 </lang>
explicit
<lang J> GCD=: +. relatively_prime=: 1 = GCD
yellowstone=: monad define
start=. #\ i. 4 + y NB. prepare minimal starting values
s=. 3 {. start NB. the sequence vector
start=. 3 }. start
while. y > # s do.
z=. {. start NB. z is the lowest number not in the sequence
while.do.
if. 0 1 -: (_2 {. s) relatively_prime z do.
if. z -.@e. s do.
break.
end.
end.
z =. >: z
end.
start=. start -. z NB. remove z from the list of starting values
s=. s , z
end.
s
) </lang>
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 load'plot' 'marker'plot yellowstone 100
Java
<lang 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>();
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);
}
} </lang>
- 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
<lang javascript>(() => {
'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();
})();</lang>
- 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
Julia
<lang 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 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)
</lang>
- 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
<lang scala>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)
}</lang>
- 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
<lang lua>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()</lang>
- 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]
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. <lang Nim>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)</lang>
- 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. <lang Nim>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()</lang>
- 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
Perl
<lang 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 $fh;</lang>
- Output:
The first 30 terms in the Yellowstone sequence: 1 2 3 4 9 8 15 14 5 6 25 12 35 16 7 10 21 20 27 22 39 11 13 33 26 45 28 51 32 17
See graph at off-site PNG image
Phix
function yellowstone(integer N) sequence a = {1, 2, 3}, b = repeat(true,3) integer i = 4 while length(a) < N do if (i>length(b) or b[i]=false) and gcd(i,a[$])=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)})
- 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}
a simple plot
include pGUI.e IupOpen() IupControlsOpen() Ihandle plot = IupPlot("MENUITEMPROPERTIES=Yes, SIZE=640x320") IupSetAttribute(plot, "TITLE", "Yellowstone Numbers"); IupSetAttribute(plot, "TITLEFONTSIZE", "10"); IupSetAttribute(plot, "TITLEFONTSTYLE", "ITALIC"); IupSetAttribute(plot, "GRIDLINESTYLE", "DOTTED"); IupSetAttribute(plot, "GRID", "YES"); IupSetAttribute(plot, "AXS_XLABEL", "n"); IupSetAttribute(plot, "AXS_YLABEL", "a(n)"); IupSetAttribute(plot, "AXS_XFONTSTYLE", "ITALIC"); IupSetAttribute(plot, "AXS_YFONTSTYLE", "ITALIC"); IupSetAttribute(plot, "AXS_YTICKSIZEAUTO", "NO"); IupSetAttribute(plot, "AXS_YTICKMAJORSIZE", "8"); IupSetAttribute(plot, "AXS_YTICKMINORSIZE", "0"); IupPlotBegin(plot) sequence y500 = yellowstone(500) for x=1 to 500 do IupPlotAdd(plot, x, y500[x]) end for {} = IupPlotEnd(plot) --IupSetAttribute(plot, "DS_MODE", "BAR") -- (optional) Ihandle dlg = IupDialog(plot,`TITLE="Yellowstone Names"`) IupShow(dlg) if platform()!=JS then IupMainLoop() IupClose() end if
Phixmonti
Require Utilitys library version 1.3
<lang Phixmonti>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 ?</lang>
- 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
<lang 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))</lang>
- 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
<lang 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</lang>
- 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
<lang python>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()</lang>
- 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>#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)))))</lang>
- 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)
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.
<lang perl6>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;</lang>
- 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
<lang rexx>/*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</lang>
- 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. <lang rexx>/*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</lang>
- output when using the input: 532
The plot is shown at three─quarter scale.
■
│
│
│
│
│
│
│
│
│
│
■ │
│ │
│ │
■ │ │
│ │ │
│ │ │
│ │ │
■ │ │ ■ │
│ │ │ ■ │ │
│ │ │ │ │ │
■ │ │ │ │ │ │
│ │ │ │ │ │ │
│ │ │ │ ■ │ │ │
■ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │
│ │ │ ■ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │
│ │ │ ■ │ │ │ │ │ │ │
│ │ ■ │ │ │ │ │ │ │ │ │
│ │ ■ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ ■
│ │ │ │ │ │ │ │ │ │ │ │ ■ ■ │ │
■ │ │ │ │ │ │ │ │ │ │ ■ ■ │ │ ■ ■ ■ │ ■ │ │ ■ │
│ ■ │ │ │ │ │ │ │ │ │ │ │ ■ │ │ ■ │ │ │ │ ■ │ ■ ■ │ │ ■ │ │ │ ■
│ │ │ │ │ │ │ │ │ │ ■ │ │ ■ ■ │ │ │ ■ ■ │ ■ ■ │ │ │ │ │ │ ■ │ │ ■ │ ■ │ │ │ │ │ │ │
■ │ │ │ │ │ │ │ │ │ │ │ │ ■ ■ │ ■ ■ │ ■ ■ │ ■ │ ■ ■ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ ■ │ │ │ │ │ │ │ │ ■ │ ■ │ │ ■ │ ■ │ │ ■ │ ■ │ │ │ │ │ ■ │ │ │ │ │ │ │ │ │ │ │ │ ■ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ ■ │ │ │ │ │ │ │ │ ■ │ ■ │ │ ■ │ ■ │ │ │ │ │ │ │ │ │ ■ ■ ■ ■ │ │ ■ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ■ │ │ │ │ │ │ │ │ │ │ │ │
│ │ ■ │ │ │ │ │ │ │ ■ │ │ ■ │ ■ ■ │ │ │ │ ■ ■ │ ■ │ ■ │ │ │ │ │ │ │ ■ ■ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ■│ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ ■ │ │ ■ │ ■ ■ │ │ ■ │ ■ ■ │ │ │ ■ ■ │ │ │ │ │ │ │ │ │ │ │ │ │ ■ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ■│ │ │ │ │■│■│ ││■│■│ │■│ │■│■│ ■│■│ ■│■│
│ │ │ │ │ ■ │ │ │ ■ ■ │ ■ │ ■ │ │ │ ■ │ │ │ │ ■ ■ │ │ │ │ │ │ ■ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ■│ │ │ │■│ │ │ │ │ │ │■│■│ ■│■│■│■│■│ ││■│■│ ■│■│││││ ││││││■│││■│││││ ││││ ││││
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│ │ │ │ │ │ ■ ■ │ ■ ■ │ ■ │ ■ │ ■ │ │ │ ■ │ ■ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │■│■│■│■│ ■│ │■│■│ ■│■│■│ ■│■│ ■│■│││││││││││││ ││││││││││■│││││││││││││││ ││││││││││ ││││││ ││││││││ ││││││││││││││││ ││││ ││││
│ ■ │ │ │ │ │ ■ ■ │ │ │ ■ │ ■ │ ■ │ ■ │ │ │ ■ │ │ │ │ │ │ ■ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │■│■│■│■│ ■│■│■│ ■│■│■│■│■│■│││││││││ ││■│││││ ││││││ ││││ ││││││││││││││││ ││││││││││││││││││││││││││ ││││││││││ ││││││ ││││││││ ││││││││││││││││ ││││ ││││
■ │ ■ │ │ │ │ ■ ■ │ ■ ■ │ ■ ■ │ │ ■ ■ │ │ │ ■ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ■ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │■│■│ │■│■│ ■│■│■│■│ ■│■│■│■│■│ ■│││││││││ ││││││ ││││││││││││││││││││ ││││││││ ││││││ ││││ ││││││││││││││││ ││││││││││││││││││││││││││ ││││││││││ ││││││ ││││││││ ││││││││││││││││ ││││ ││││
│ │ │ │ │ ■ │ ■ ■ │ ■ ■ ■ ■ │ │ ■ │ ■ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │■│■│■│■│■│ ■│■│ ■│■│■│■│ ■│■│■│││││■│││││ ││││││││ ││││││││││ ││││││││││ ││││││ ││││││││││││││││││││ ││││││││ ││││││ ││││ ││││││││││││││││ ││││││││││││││││││││││││││ ││││││││││ ││││││ ││││││││ ││││││││││││││││ ││││ ││││
│ ■ │ │ │ │ ■ ■ ■ ■ │ │ ■ │ │ ■ │ │ ■ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ■ ■│ │ │■│ │ │■│■│■│■│ ■│■│ ■│■│■│■│■│■│││││││││││ ││││ ││││││││ ││││││││││││││││ ││││││││ ││││││││││ ││││││││││ ││││││ ││││││││││││││││││││ ││││││││ ││││││ ││││ ││││││││││││││││ ││││││││││││││││││││││││││ ││││││││││ ││││││ ││││││││ ││││││││││││││││ ││││ ││││
│ │ │ │ ■ │ ■ ■ ■ │ ■ ■ ■ │ ■ │ │ │ │ │ │ │ │ │ │ │ │ │ │ ■ │ │ │ │ │ │ │ │ │■│ │■│■│■│■│■│■│■│■│■│■│ ││■│■│││ ■│■│││││││││ ││││ ││││││││││││││││││││││ ││││ ││││││││ ││││││││││││││││ ││││││││ ││││││││││ ││││││││││ ││││││ ││││││││││││││││││││ ││││││││ ││││││ ││││ ││││││││││││││││ ││││││││││││││││││││││││││ ││││││││││ ││││││ ││││││││■││││││││││││││││■││││■││││
│ ■ │ │ │ ■ ■ │ ■ ■ ■ │ ■ ■ ■ │ │ │ │ ■ │ │ ■ │ │ │ │ │ │ │ │ │ │ │ │ │■│■│ │■│■│■│■│ ■│■│ ■│■│■│■│ ■│││■│││││││││││││││││││││ ││││││││ ││││││││││││ ││││ ││││││││││││││││││││││ ││││ ││││││││ ││││││││││││││││ ││││││││ ││││││││││ ││││││││││ ││││││ ││││││││││││││││││││ ││││││││ ││││││ ││││ ││││││││││││││││■││││││││││││││││││││││││││■││││││││││■││││││■│││││││││││││││││││││││││││││││││││
■ │ │ │ ■ ■ ■ ■ │ ■ ■ ■ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │■│■│ ■│■│■│■│■ ■│■│■│■│■│ ■│││││ ■│││││││││ ││││ ││││││││ ││││││││││││││││││││││││││ ││││││││ ││││││││││││ ││││ ││││││││││││││││││││││ ││││ ││││││││ ││││││││││││││││ ││││││││ ││││││││││ ││││││││││■││││││■││││││││││││││││││││■││││││││■││││││■││││■│││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││
■ │ ■ ■ │ │ ■ ■ ■ │ │ │ │ │ ■ │ │ │ │ │ │ │ │ │ │■│ │■│■│■│ ■│■│■│■ ■│■│■│■│■│■■■│││││ │││││││││ ││││││││││ ││││││ ││││││││││ ││││ ││││││││ ││││││││││││││││││││││││││ ││││││││ ││││││││││││ ││││ ││││││││││││││││││││││■││││■││││││││■││││││││││││││││■││││││││■││││││││││■│││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││
■ ■ │ │ ■ ■ ■ ■ ■ ■ │ │ │ │ │ ■ │ │ │ │ │ ■ ■■│ │■│ ■│■│■│■│■ ■│■│■│■│■│ ■│││ ■│││││││ │││││││ ││││││││││││││││││ │││││││││ ││││││││││ ││││││ ││││││││││ ││││ ││││││││ ││││││││││││││││││││││││││■││││││││■││││││││││││■││││■│││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││
■ ■ ■ ■ ■ │ │ │ │ ■ ■ │ │ │■│■│ │ │■│ ■│■│■│■│■│■│ ■│■│■│■│■│││ ■│││ │││││││││ ││││││││││ ││││ ││││││││ │││││││ ││││││││││││││││││■│││││││││■││││││││││■││││││■││││││││││■││││■││││││││■│││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││
■ ■ │ ■ │ ■ │ ■ ■ ■■ │■│■■ │■│■│■│■│ ■│■│■│■ ■│││││■│■│││ ││││││││││││ ││││││││││││■││││■│││││││││■││││││││││■││││■││││││││■│││││││■│││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││││
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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
<lang 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
</lang>
- 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...
Ruby
<lang 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)</lang>
- 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
<lang 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),
}
}</lang>
- 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
See: yellowstone.png (offsite PNG image)
Tcl
<lang Tcl>proc gcd {a b} {
while {$b} {
lassign [list $b [expr {$a % $b}]] a b
}
return $a
}
proc gen_yellowstones Template:MaxN 30 {
set r {}
for {set n 1} {$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]</lang>
- 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
VBA
<lang Tcl> 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 </lang>
- 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
Wren
Without the extra credit part. <lang ecmascript>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)</lang>
- 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]
zkl
This sequence is limited to the max size of a Dictionary, 64k <lang zkl>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);
}</lang> <lang zkl>println("The first 30 entries of the Yellowstone permutation:"); yellowstoneW().walk(30).concat(", ").println();</lang>
- 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 <lang zkl>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();</lang> Offsite Image: yellowstone
