A172517 Number of ways to place 2 nonattacking queens on an n X n toroidal board.

0, 0, 0, 32, 100, 288, 588, 1152, 1944, 3200, 4840, 7200, 10140, 14112, 18900, 25088, 32368, 41472, 51984, 64800, 79380, 96800, 116380, 139392, 165000, 194688, 227448, 264992, 306124, 352800, 403620, 460800, 522720, 591872, 666400, 749088, 837828

Offset: 1

Author

Vaclav Kotesovec, Feb 05 2010

Keywords

• nonn
• nice
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• V. Kotesovec, Number of ways of placing non-attacking queens and kings on boards of various sizes
• Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Crossrefs

Cf. A007705, A036464.

Programs

Mathematica

CoefficientList[Series[- 4 x^3 (x^3 + 6 x^2 + 9 x + 8) / ((x - 1)^5 (x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, May 29 2013 *)
LinearRecurrence[{2,2,-6,0,6,-2,-2,1},{0,0,0,32,100,288,588,1152},40] (* Harvey P. Dale, Sep 22 2015 *)

Formula

a(n) = n^2*(n-2)^2/2 if n is even and a(n) = n^2*(n-1)(n-3)/2 if n is odd.

G.f.: -4*x^4*(x^3+6*x^2+9*x+8) / ((x-1)^5*(x+1)^3). - Colin Barker, Jan 09 2013

a(n) = 2*a(n-1)+2*a(n-2)-6*a(n-3)+6*a(n-5)-2*a(n-6)-2*a(n-7)+a(n-8). - Wesley Ivan Hurt, May 28 2021