A177755 Number of ways to place 2 nonattacking bishops on an n X n toroidal board.

0, 4, 18, 80, 200, 468, 882, 1600, 2592, 4100, 6050, 8784, 12168, 16660, 22050, 28928, 36992, 46980, 58482, 72400, 88200, 106964, 128018, 152640, 180000

Offset: 1

Author

Vaclav Kotesovec, May 13 2010

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• V. Kotesovec, Non-attacking chess pieces, 6ed, 2013
• Index entries for linear recurrences with constant coefficients, signature (2, 2, -6, 0, 6, -2, -2, 1).

Crossrefs

Cf. A172123.

Programs

Mathematica

Table[(n^2 (2n^2-4n+3+(-1)^n))/4,{n,30}] (* or *) LinearRecurrence[ {2,2,-6,0,6,-2,-2,1},{0,4,18,80,200,468,882,1600},30] (* Harvey P. Dale, Mar 06 2013 *)
CoefficientList[Series[- 2 x (x^5 + 8 x^4 + 14 x^3 + 18 x^2 + 5 x + 2) / ((x - 1)^5 (x + 1)^3), {x, 0, 50}], x] (* Vincenzo Librandi, May 31 2013 *)

Formula

Explicit formula: 1/4*n^2*(2*n^2-4*n+3+(-1)^n).

G.f.: -2*x^2*(x^5+8*x^4+14*x^3+18*x^2+5*x+2)/((x-1)^5*(x+1)^3).

a(1)=0, a(2)=4, a(3)=18, a(4)=80, a(5)=200, a(6)=468, a(7)=882, a(8)=1600, 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). - Harvey P. Dale, Mar 06 2013