A202657 Number of ways to place 6 nonattacking semi-queens on an n X n board.

0, 0, 0, 0, 0, 83, 6107, 126376, 1377328, 9984758, 54399330, 239675936, 895773148, 2935757573, 8641608781, 23259768860, 58039719112, 135720432200, 299995484600, 631220344328, 1271607596876, 2464466665667, 4613731163831, 8372196591052, 14769606793684, 25395151577010

Offset

1,6

Comments

Two semi-queens do not attack each other if they are in the same northwest-southeast diagonal.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Christopher R. H. Hanusa, Thomas Zaslavsky, A q-queens problem. VII. Combinatorial types of nonattacking chess riders, arXiv:1906.08981 [math.CO], 2019.

V. Kotesovec, Non-attacking chess pieces

Formula

a(n) = n^12/720 - n^11/18 + 73*n^10/72 - 72247*n^9/6480 + 5909*n^8/72 - 320653*n^7/756 + 112795*n^6/72 - 8892919*n^5/2160 + 8086231*n^4/1080 - 5740271*n^3/648 + 2598425*n^2/432 - 13161367*n/7560 + (n^5/4 - 77*n^4/12 + 757*n^3/12 - 7007*n^2/24 + 14581*n/24 - 1677/4)*floor(n/2) + (64*n/9 - 88/9)*floor(n/3) + (8*n/3 - 52/9)*floor((n + 1)/3).

G.f.: -x^6*(31709*x^16 + 377288*x^15 + 2265487*x^14 + 8441426*x^13 + 22166758*x^12 + 43217858*x^11 + 64805639*x^10 + 75943200*x^9 + 70077016*x^8 + 50738668*x^7 + 28477437*x^6 + 12074418*x^5 + 3711058*x^4 + 771370*x^3 + 96173*x^2 + 5692*x + 83)/((x-1)^13*(x+1)^6*(x^2+x+1)^2).

Mathematica

Rest@ CoefficientList[Series[-x^6*(31709 x^16 + 377288 x^15 + 2265487 x^14 + 8441426 x^13 + 22166758 x^12 + 43217858 x^11 + 64805639 x^10 + 75943200 x^9 + 70077016 x^8 + 50738668 x^7 + 28477437 x^6 + 12074418 x^5 + 3711058 x^4 + 771370 x^3 + 96173 x^2 + 5692 x + 83)/((x - 1)^13*(x + 1)^6*(x^2 + x + 1)^2), {x, 0, 26}], x] (* Michael De Vlieger, Aug 19 2019 *)

Crossrefs

Cf. A099152, A176186, A103220, A202654, A202655, A202656.

Sequence in context: A017746 A220649 A252812 * A180846 A097839 A268987

Adjacent sequences: A202654 A202655 A202656 * A202658 A202659 A202660

Keyword

nonn

Author

Vaclav Kotesovec, Dec 22 2011

Status

approved