A035288 Number of ways to place a non-attacking white and black bishop on n X n chessboard.

0, 8, 52, 184, 480, 1040, 1988, 3472, 5664, 8760, 12980, 18568, 25792, 34944, 46340, 60320, 77248, 97512, 121524, 149720, 182560, 220528, 264132, 313904, 370400, 434200, 505908, 586152, 675584, 774880, 884740, 1005888, 1139072, 1285064, 1444660, 1618680, 1807968

Offset: 1

Author

Erich Friedman

Keywords

• nonn
• easy

Examples

			There are 52 ways of putting 2 distinct bishops on 3 X 3 so that neither can capture the other.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A172123.

Programs

Magma

[(3*n^4-4*n^3+3*n^2-2*n)/3: n in [1..35]]; // Vincenzo Librandi, May 04 2013

Mathematica

Table[(3n^4-4n^3+3n^2-2n)/3, {n,40}] (* or *) LinearRecurrence[ {5,-10,10,-5,1}, {0,8,52,184,480}, 40] (* Harvey P. Dale, Nov 19 2011 *)

PARI

a(n)=(3*n^4-4*n^3+3*n^2-2*n)/3; \\ Joerg Arndt, May 04 2013

Formula

a(n) = (3*n^4 - 4*n^3 + 3*n^2 - 2*n)/3.

a(n) = 2 * A172123(n). - Vaclav Kotesovec, Nov 28 2011

a(n) = 5*a(n-1) -10*a(n-2) +10*a(n-3) -5*a(n-4) +a(n-5). - Harvey P. Dale, Nov 19 2011

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

E.g.f.: exp(x)*x^2*(12 + 14*x + 3*x^2)/3. - Stefano Spezia, Aug 21 2025