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A274640 Counterclockwise square spiral constructed by greedy algorithm, so that each row, column, and diagonal contains distinct numbers. 40
1, 2, 3, 4, 2, 3, 4, 5, 6, 1, 4, 6, 2, 1, 6, 5, 3, 1, 5, 2, 6, 1, 2, 4, 5, 3, 7, 8, 5, 4, 9, 7, 8, 3, 10, 11, 4, 7, 8, 6, 3, 9, 5, 7, 8, 9, 10, 11, 12, 6, 8, 9, 11, 10, 12, 13, 7, 6, 10, 9, 12, 13, 14, 15, 8, 2, 9, 12, 7, 10, 11, 13, 14, 10, 9, 6, 13, 5, 3, 15, 16, 7, 1, 10, 13, 12, 14, 11, 15, 3, 8, 5, 1, 12, 11, 14, 7, 4, 2, 16, 9, 17, 1, 8, 11 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Presumably every row, column, and diagonal is a permutation of the natural numbers, but is there a proof? - N. J. A. Sloane, Jul 10 2016

The n-th cell in the spiral has coordinates x = A174344(n+1), y = A274923(n+1). - N. J. A. Sloane, Jul 11 2016

From Robert G. Wilson v, Dec 25 2016: (Start) [Memo: all these numbers need to decreased by 1, since the offset here is 0. See A324481. - N. J. A. Sloane, Jul 23 2017. Furthermore, the numbers don't seem correct, even after subtracting 1. - N. J. A. Sloane, Jul 04 2019]

Index of first appearance of k = 1,2,3,...: 1, 2, 3, 7, 8, 15, 17, 25, 35, 41, 47, 61, 62, 89, 98, 99, 121, 129, 130, 143, 197, 208, 225, 239, 271, ..., .

1 appears at: 1, 4, 12, 19, 22, 33, 42, 68, 79, 120, 179, 194, 302, 311, 445, 489, 511, 558, 630, 708, 847, 877, 907, ..., .

2 appears at: 2, 5, 9, 16, 48, 52, 70, 73, 88, 95, 110, 146, 280, 291, 309, 327, 488, 605, 656, 681, 735, 778, 1000, ..., .

3 appears at: 3, 6, 10, 23, 29, 36, 56, 76, 97, 105, 153, 168, 184, 252, 338, 437, 457, 670, 818, 906, 953, 967, ..., . (End).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..20000

F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, arXiv:1907.09120, July 2019

Alois P. Heinz, Distribution of a(n) for n <= 4010000

Kerry Mitchell, Color-coded version of spiral, (1): the colors represent the values, from black (small) to white (large) (jpg file, low resolution)

Kerry Mitchell, Color-coded version of spiral, (1a): the colors represent the values, from black (small) to white (large) (tiff file, much higher resolution)

Kerry Mitchell, Color-coded version of spiral, (2): values <= 100 are black and those > 100 are white.

Zak Seidov, Distribution of a(n) for first 20001 terms

EXAMPLE

The spiral begins:

.

   9--16---2---4---7--14--11--12---1---5---8

   |                                       |

  17   8--15--14--13--12---9--10---6---7   3

   |   |                               |   |

   1   2   4--11--10---3---8---7---9  13  15

   |   |   |                       |   |   |

   8   9   7   3---5---6---1---2   4  12  11

   |   |   |   |               |   |   |   |

  11  12   8   1   2---4---3   6   5  10  14

   |   |   |   |   |       |   |   |   |   |

  15   7   6   5   3   1---2   4   8  11  12

   |   |   |   |   |           |   |   |   |

  14  10   3   2   4---5---6---1   7   9  13

   |   |   |   |                   |   |   |

   7  11   9   6---1---2---4---5---3   8  10

   |   |   |                           |   |

   4  13   5---7---8---9--10--11--12---6   1

   |   |                                   |

  12  14--10---9---6--13---5---3--15--16---7

   |

  10--15---1--12--16---8--14--13--11--18--17

.

The 8 spokes (A274924-A274931) begin:

E:  1, 2, 4,  8, 11, 12, 16,  9, 19, 24, 22, ...

NE: 1, 3, 2,  9,  7,  8, 12, 15, 13, 17, 20, ...

N:  1, 4, 6,  3, 12, 14, 15, 18, 20, 26, 25, ...

NW: 1, 2, 3,  4,  8,  9,  7, 11, 14, 10, 22, ...

W:  1, 3, 5,  6,  7, 15, 10, 17, 13, 25, 14, ...

SW: 1, 4, 6,  5, 14, 10, 11, 23, 16, 18, 21, ...

S:  1, 5, 2,  9, 13,  8,  7, 11, 10, 17, 19, ...

SE: 1, 6, 5, 12, 16, 17, 21, 24, 27, 13, 15, ...

MAPLE

#  Maple program from Alois P. Heinz, Jul 12 2016:

fx:= proc(n) option remember; `if`(n=1, 0, (k->

       fx(n-1)+sin(k*Pi/2))(floor(sqrt(4*(n-2)+1)) mod 4))

     end:

fy:= proc(n) option remember; `if`(n=1, 0, (k->

       fy(n-1)-cos(k*Pi/2))(floor(sqrt(4*(n-2)+1)) mod 4))

     end:

b:= proc() 0 end:

a:= proc(n) local x, y, s, i, t, m;

      x, y:= fx(n+1), fy(n+1);

      if b(x, y) > 0 then b(x, y)

    else s:={};

    for i do t:=b(x+i, y+i); if t>0 then s:=s union {t} else break fi od;

    for i do t:=b(x-i, y-i); if t>0 then s:=s union {t} else break fi od;

    for i do t:=b(x+i, y-i); if t>0 then s:=s union {t} else break fi od;

    for i do t:=b(x-i, y+i); if t>0 then s:=s union {t} else break fi od;

    for i do t:=b(x+i, y  ); if t>0 then s:=s union {t} else break fi od;

    for i do t:=b(x-i, y  ); if t>0 then s:=s union {t} else break fi od;

    for i do t:=b(x  , y+i); if t>0 then s:=s union {t} else break fi od;

    for i do t:=b(x  , y-i); if t>0 then s:=s union {t} else break fi od;

         for m while m in s do od;

         b(x, y):= m

      fi

    end:

seq(a(n), n=0..1000);

MATHEMATICA

fx[n_] := fx[n] = If[n == 1, 0, Function[k, fx[n-1] + Sin[k*Pi/2]][Mod[Floor[Sqrt[4*(n-2)+1]], 4]]]; fy[n_] := fy[n] = If[n == 1, 0, Function[k, fy[n-1] - Cos[k*Pi/2]][Mod[Floor[Sqrt[4*(n-2)+1]], 4]]]; Clear[b]; b[_, _] = 0; a[n_] := Module[{x, y, s, i, t, m}, {x, y} = {fx[n+1], fy[n+1]}; If[b[x, y] > 0, b[x, y], s = {};

For[i=1, True, i++, t=b[x+i, y+i]; If[t>0, s=Union[s, {t}], Break[]]];

For[i=1, True, i++, t=b[x-i, y-i]; If[t>0, s=Union[s, {t}], Break[]]];

For[i=1, True, i++, t=b[x+i, y-i]; If[t>0, s=Union[s, {t}], Break[]]];

For[i=1, True, i++, t=b[x-i, y+i]; If[t>0, s=Union[s, {t}], Break[]]];

For[i=1, True, i++, t=b[x+i, y  ]; If[t>0, s=Union[s, {t}], Break[]]];

For[i=1, True, i++, t=b[x-i, y  ]; If[t>0, s=Union[s, {t}], Break[]]];

For[i=1, True, i++, t=b[x  , y+i]; If[t>0, s=Union[s, {t}], Break[]]];

For[i=1, True, i++, t=b[x  , y-i]; If[t>0, s=Union[s, {t}], Break[]]];

m = 1; While[MemberQ[s, m], m++]; b[x, y] = m]]; Table[a[n], {n, 0, 1000}] (* Jean-Fran├žois Alcover, Nov 14 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A274641 (the same spiral, but starting with 0 not 1), A174344, A274923.

The 8 spokes are A274924-A274931.

The East-West axis is A275877 (see also A324680), the North-South axis is A276036.

Positions of 1's and 2's give A273059 and A275116.

In the same spirit as the infinite Sudoku array A269526.

Cf. A324481 (position of first n).

Sequence in context: A173524 A049865 A070771 * A245341 A215088 A182985

Adjacent sequences:  A274637 A274638 A274639 * A274641 A274642 A274643

KEYWORD

nonn,nice

AUTHOR

Zak Seidov and Kerry Mitchell, Jun 30 2016

EXTENSIONS

Corrected and extended by Alois P. Heinz, Jul 12 2016

STATUS

approved

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Last modified September 20 16:30 EDT 2019. Contains 327242 sequences. (Running on oeis4.)