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

1,4

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).

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

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 *)

Crossrefs

Cf. A007705, A036464.

Sequence in context: A192293 A188862 A228686 * A194645 A134845 A167982

Adjacent sequences: A172514 A172515 A172516 * A172518 A172519 A172520

Keyword

nonn,nice,easy

Author

Vaclav Kotesovec, Feb 05 2010

Status

approved