A194651 Number of ways to place 3 nonattacking kings on an n X n cylindrical chessboard.

0, 0, 0, 88, 785, 3528, 11151, 28560, 63513, 127520, 236863, 413736, 687505, 1096088, 1687455, 2521248, 3670521, 5223600, 7286063, 9982840, 13460433, 17889256, 23466095, 30416688, 38998425, 49503168, 62260191, 77639240, 96053713, 117963960, 143880703

Offset

1,4

Links

Table of n, a(n) for n=1..31.

Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 163.

Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Formula

a(n) = 1/6*n*(n^5 - 27*n^3 + 18*n^2 + 194*n - 228), n>=4.

G.f.: -x^4*(15*x^6 - 89*x^5 + 196*x^4 - 140*x^3 - 119*x^2 + 169*x + 88)/(x-1)^7.

Mathematica

CoefficientList[Series[-x^3*(15*x^6 - 89*x^5 + 196*x^4 - 140*x^3 - 119*x^2 + 169*x + 88)/(x - 1)^7, {x, 0, 30}], x] (* Wesley Ivan Hurt, Dec 27 2023 *)

Prog

(PARI) a(n)=if(n>3, n*(n^5-27*n^3+18*n^2+194*n-228)/6, 0) \\ Charles R Greathouse IV, May 28 2026

Crossrefs

Cf. A061996, A179404, A194650.

Sequence in context: A183839 A297228 A137124 * A072432 A136951 A136933

Adjacent sequences: A194648 A194649 A194650 * A194652 A194653 A194654

Keyword

nonn,easy

Author

Vaclav Kotesovec, Aug 31 2011

Status

approved