A172518 Number of ways to place 3 nonattacking queens on an n X n toroidal board.

0, 0, 0, 0, 100, 576, 2156, 7168, 17496, 41600, 82280, 161280, 280540, 486080, 774900, 1232896, 1844976, 2757888, 3933456, 5606400, 7699860, 10570560, 14081980, 18754560, 24365000, 31647616, 40258296, 51204608, 63979916

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 (2,4,-10,-5,20,0,-20,5,10,-4,-2,1).

Crossrefs

Cf. A047659, A007705, A172517.

Programs

Mathematica

CoefficientList[Series[- 4 x^4 (9 x^6 + 94 x^5 + 199 x^4 + 388 x^3 + 151 x^2 + 94 x + 25) / ((x - 1)^7 (x + 1)^5), {x, 0, 50}], x] (* Vincenzo Librandi, May 29 2013 *)
LinearRecurrence[{2,4,-10,-5,20,0,-20,5,10,-4,-2,1},{0,0,0,0,100,576,2156,7168,17496,41600,82280,161280},30] (* Harvey P. Dale, Dec 27 2014 *)

Formula

a(n) = n^2*(n-2)*(n-4)*(n^2-6*n+12)/6 if n is even and a(n) = n^2*(n-1)*(n-3)*(n^2-8*n+18)/6 if n is odd. - Vaclav Kotesovec, Jan 31 2010

G.f.: -4*x^5*(9*x^6+94*x^5+199*x^4+388*x^3+151*x^2+94*x+25) / ((x-1)^7*(x+1)^5). - Colin Barker, Jan 09 2013