A194650 Number of ways to place 2 nonattacking kings on an n X n cylindrical chessboard.

0, 0, 9, 68, 215, 504, 1001, 1784, 2943, 4580, 6809, 9756, 13559, 18368, 24345, 31664, 40511, 51084, 63593, 78260, 95319, 115016, 137609, 163368, 192575, 225524, 262521, 303884, 349943, 401040, 457529, 519776, 588159, 663068, 744905, 834084, 931031, 1036184

Offset

1,3

Links

Table of n, a(n) for n=1..38.

V. Kotesovec, Number of ways of placing non-attacking queens, kings, bishops and knights

Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Formula

a(n) = 1/2*n*(n^3 - 9*n + 6), n>=3.

G.f.: x^3*(4*x^4 - 19*x^3 + 35*x^2 - 23*x - 9)/(x-1)^5.

Mathematica

CoefficientList[Series[x^2*(4*x^4 - 19*x^3 + 35*x^2 - 23*x - 9)/(x - 1)^5, {x, 0, 50}], x] (* Wesley Ivan Hurt, Dec 27 2023 *)

Prog

(PARI) a(n)=if(n>2, n*(n^3-9*n+6)/2, 0) \\ Charles R Greathouse IV, May 26 2026

Crossrefs

Cf. A061995, A179403.

Sequence in context: A002051 A231192 A133120 * A048742 A121633 A091708

Adjacent sequences: A194647 A194648 A194649 * A194651 A194652 A194653

Keyword

nonn,easy

Author

Vaclav Kotesovec, Aug 31 2011

Status

approved