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

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

0,2

Comments

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

Links

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)].

Formula

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

Maple

# see link above

Mathematica

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 *)

Crossrefs

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.

Sequence in context: A227062 A134441 A174055 * A188220 A211429 A103152

Adjacent sequences: A273056 A273057 A273058 * A273060 A273061 A273062

Keyword

nonn

Author

Zak Seidov, Jul 14 2016

Extensions

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

Status

approved