A274640 -id:A274640 - cp's OEIS frontend

A274641 Counterclockwise square spiral constructed by greedy algorithm, so that each row, column, and diagonal contains distinct numbers. Start with 0 (so in this version a(n) = A274640(n) - 1).

0, 1, 2, 3, 1, 2, 3, 4, 5, 0, 3, 5, 1, 0, 5, 4, 2, 0, 4, 1, 5, 0, 1, 3, 4, 2, 6, 7, 4, 3, 8, 6, 7, 2, 9, 10, 3, 6, 7, 5, 2, 8, 4, 6, 7, 8, 9, 10, 11, 5, 7, 8, 10, 9, 11, 12, 6, 5, 9, 8, 11, 12, 13, 14, 7, 1, 8, 11, 6, 9, 10, 12, 13, 9, 8, 5, 12, 4, 2, 14, 15, 6, 0, 9, 12, 11, 13, 10, 14, 2, 7, 4, 0, 11, 10, 13, 6, 3, 1, 15, 8, 16, 0, 7, 10

Offset: 0

N. J. A. Sloane, Jul 09 2016, based on the entry A274640 from Zak Seidov and Kerry Mitchell, Jun 30 2016

nonn

See A274640 for further information.

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

			From _Jon E. Schoenfield_, Dec 26 2016: (Start)
The spiral begins:
.
   8--15---1---3---6--13--10--11---0---4---7
   |                                       |
  16   7--14--13--12--11---8---9---5---6   2
   |   |                               |   |
   0   1   3--10---9---2---7---6---8  12  14
   |   |   |                       |   |   |
   7   8   6   2---4---5---0---1   3  11  10
   |   |   |   |               |   |   |   |
  10  11   7   0   1---3---2   5   4   9  13
   |   |   |   |   |       |   |   |   |   |
  14   6   5   4   2   0---1   3   7  10  11
   |   |   |   |   |           |   |   |   |
  13   9   2   1   3---4---5---0   6   8  12
   |   |   |   |                   |   |   |
   6  10   8   5---0---1---3---4---2   7   9
   |   |   |                           |   |
   3  12   4---6---7---8---9--10--11---5   0
   |   |                                   |
  11  13---9---8---5--12---4---2--14--15---6
   |
   9--14---0--11--15---7--13--12--10--17--16
.
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..20000 [Based on Alois Heinz's b-file for A274640]
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Rémy Sigrist, Colored representation of the spiral for -500 <= x, y <= 500
N. J. A. Sloane, Confessions of a Sequence Addict (AofA2017), slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.

Cf. A274640 (if start with 1 at center), A324481 (position of first n).

For the eight spokes see A324774-A324781.

A274924 East spoke of spiral in A274640.

Original entry on oeis.org

1, 2, 4, 8, 11, 12, 16, 9, 19, 24, 22, 18, 27, 26, 21, 37, 43, 39, 40, 49, 28, 29, 32, 46, 55, 60, 45, 48, 66, 73, 70, 76, 83, 77, 65, 75, 42, 62, 94, 96, 101, 103, 67, 63, 112, 80, 113, 58, 107, 64, 108, 120, 109, 69, 124, 130, 140, 134, 122, 133, 139, 129

Offset: 0

N. J. A. Sloane, Jul 12 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

Cf. A274640, A274641. The 8 spokes are A274924-A274931.

More terms from Alois P. Heinz, Jul 12 2016

A274931 Southeast spoke of spiral in A274640.

Original entry on oeis.org

1, 6, 5, 12, 16, 17, 21, 24, 27, 13, 15, 36, 40, 43, 47, 54, 23, 56, 59, 62, 39, 68, 69, 35, 46, 37, 42, 90, 50, 97, 94, 99, 102, 57, 107, 115, 117, 116, 120, 66, 130, 131, 73, 77, 140, 143, 76, 80, 78, 159, 161, 156, 165, 92, 91, 174, 89, 181, 104, 187, 190

Offset: 0

N. J. A. Sloane, Jul 12 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

Cf. A274640, A274641. The 8 spokes are A274924-A274931.

More terms from Alois P. Heinz, Jul 12 2016

A273059 Positions of 1's in A274640: Greedy Queens on a spiral. Equivalently, positions of 0's in A274641.

Original entry on oeis.org

0, 9, 13, 17, 21, 82, 92, 102, 112, 228, 244, 260, 276, 445, 467, 489, 511, 630, 656, 682, 708, 967, 999, 1031, 1063, 1377, 1415, 1453, 1491, 1858, 1902, 1946, 1990, 2411, 2461, 2511, 2561, 3037, 3093, 3149, 3205, 3734, 3796, 3858, 3920, 4239, 4305, 4371, 4437, 5056, 5128, 5200, 5272, 5946

Offset: 0

Zak Seidov, Jul 14 2016

nonn

What is the reason for the three "lines" in the graph of first differences (see link, also A275915)?
Apparently they are related to the fact that "ones" are concentrated along two main diagonals of the spiral A274640, see the graph "Spiral A274640 with ones shown".
This is the Greedy Queens construction on a spiral (cf. A065188). Follow a counterclockwise spiral starting at the origin, and place a queen iff it is not attacked by any existing queen. This same problem is described in a different but equivalent way in A140100 and A140101. See A140101 for a conjectured recurrence which underlies all these sequences. - N. J. A. Sloane, Aug 28-30, 2016

Alois P. Heinz, Table of n, a(n) for n = 0..30288 (First 101 terms from Zak Seidov)
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
Alois P. Heinz, Maple program for A273059
Alois P. Heinz, Positions of first 1409 1's in plane of A274640 (Equivalently, positions of first 1409 0's in plane of A274641.)
Zak Seidov, Graph of first differences of A273059.
Zak Seidov, Spiral A274640 with ones shown.
N. J. A. Sloane, For each of the first 1409 0's in A274641, list [n, x(n), y(n)].

Cf. A274640, A065188, A275915 (first differences).
The four spokes are A275916, A275917, A275918, A275919.
A140100 and A140101 describe this same problem in a different way.

Maple
```
# see link above
```

fx[n_] := fx[n] = If[n == 1, 0, fx[n - 1] + Sin[#*Pi/2]& @ Mod[Floor[Sqrt[ 4*(n - 2) + 1]], 4]];
fy[n_] := fy[n] = If[n == 1, 0, fy[n - 1] - Cos[k*Pi/2]& @ Mod[Floor[Sqrt[ 4*(n - 2) + 1]], 4]];
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]];
Flatten[Position[a /@ Range[0, 10^4], 1]] - 1 (* Jean-François Alcover, Feb 25 2020, after Alois P. Heinz *)

A274640(a(n)) = 1 (this is simply a restatement of the definition).

Offset changed to 0 by N. J. A. Sloane, Aug 31 2016

A275172 Positions of 3 in A274640.

Original entry on oeis.org

2, 5, 16, 25, 33, 40, 78, 89, 158, 172, 221, 237, 376, 395, 415, 523, 570, 749, 777, 790, 876, 923, 1016, 1114, 1182, 1303, 1341, 1473, 1512, 1754, 1839, 1970, 2105, 2267, 2315, 2488, 2538, 2957, 3012, 3204, 3262, 3467, 3527, 3646, 3983, 4110, 4276, 4744, 4815

Offset: 1

Zak Seidov, Jul 19 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10074
Alois P. Heinz, Positions of first 1411 3's in plane of A274640

Cf. A274640, A273059, A275116, A275188, A275189, A275190, A275191, A275192, A275193, A275194.

A274640(a(n)) = 3.

A275188 Positions of 8 in A274640.

Original entry on oeis.org

27, 32, 38, 44, 50, 64, 90, 103, 115, 125, 160, 181, 187, 207, 225, 325, 381, 669, 778, 806, 906, 1096, 1217, 1490, 1551, 1691, 1732, 1817, 1901, 2036, 2172, 2244, 2313, 2388, 2537, 2590, 2664, 2745, 2770, 2956, 3038, 3293, 3735, 3797, 3857, 4080, 4210, 4274

Offset: 1

Zak Seidov, Jul 19 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10074
Alois P. Heinz, Positions of first 1412 8's in plane of A274640

Cf. A274640, A275172, A275189, A275190, A275191, A275192, A275193, A275194.

A274640(a(n)) = 8.

A275189 Positions of 4 in A274640.

Original entry on oeis.org

3, 6, 10, 23, 29, 36, 97, 108, 185, 199, 252, 310, 337, 404, 424, 446, 618, 631, 707, 833, 848, 905, 1000, 1097, 1196, 1253, 1269, 1378, 1511, 1552, 1796, 1815, 2014, 2060, 2363, 2589, 2640, 2692, 2928, 3179, 3261, 3554, 3673, 3768, 3829, 4370, 4504, 4707

Offset: 1

Zak Seidov, Jul 19 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10074
Alois P. Heinz, Positions of first 1410 4's in plane of A274640

Cf. A274640, A275172, A275188, A275190, A275191, A275192, A275193, A275194.

A274640(a(n))=4.

A275190 Positions of 5 in A274640.

Original entry on oeis.org

7, 15, 18, 24, 28, 42, 77, 91, 142, 227, 261, 275, 356, 374, 384, 479, 501, 670, 735, 763, 791, 863, 1050, 1130, 1164, 1288, 1492, 1631, 1648, 1713, 1838, 2015, 2034, 2268, 2413, 2562, 2611, 2715, 2767, 3036, 3148, 3263, 3319, 3954, 4046, 4145, 4541, 4676

Offset: 1

Zak Seidov, Jul 19 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10075
Alois P. Heinz, Positions of first 1410 5's in plane of A274640

Cf. A274640, A275172, A275188, A275189, A275191, A275192, A275193, A275194.

A274640(a(n)) = 5.

A275191 Positions of 6 in A274640.

Original entry on oeis.org

8, 11, 14, 20, 39, 49, 57, 75, 145, 178, 222, 292, 319, 416, 425, 546, 655, 721, 734, 862, 892, 924, 1030, 1113, 1129, 1287, 1342, 1414, 1608, 1797, 1859, 1947, 2104, 2125, 2173, 2342, 2462, 2691, 2714, 2983, 3236, 3408, 3494, 3955, 4081, 4240, 4474, 4677

Offset: 1

Zak Seidov, Jul 19 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10074
Alois P. Heinz, Positions of first 1410 6's in plane of A274640

Cf. A274640, A275172, A275188, A275189, A275190, A275192, A275193, A275194.

A274640(a(n)) = 6.

A275192 Positions of 7 in A274640.

Original entry on oeis.org

26, 31, 37, 43, 56, 68, 81, 96, 107, 127, 144, 162, 208, 247, 304, 358, 363, 509, 579, 894, 1048, 1080, 1115, 1398, 1590, 1630, 1671, 1731, 1882, 1925, 2151, 2198, 2245, 2612, 2743, 2768, 2849, 2902, 2927, 3011, 3320, 3436, 3493, 3645, 3706, 4017, 4341, 4572

Offset: 1

Zak Seidov, Jul 19 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..10073
Alois P. Heinz, Positions of first 1411 7's in plane of A274640

Cf. A274640, A275172, A275188, A275189, A275190, A275191, A275193, A275194.

A274640(a(n)) = 7.

cp's OEIS Frontend

A274641 Counterclockwise square spiral constructed by greedy algorithm, so that each row, column, and diagonal contains distinct numbers. Start with 0 (so in this version a(n) = A274640(n) - 1).

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A274924 East spoke of spiral in A274640.

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A274931 Southeast spoke of spiral in A274640.

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A273059 Positions of 1's in A274640: Greedy Queens on a spiral. Equivalently, positions of 0's in A274641.

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A275172 Positions of 3 in A274640.

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A275188 Positions of 8 in A274640.

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A275189 Positions of 4 in A274640.

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A275190 Positions of 5 in A274640.

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A275191 Positions of 6 in A274640.

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A275192 Positions of 7 in A274640.

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