A179404 Number of ways to place 3 nonattacking kings on an n X n toroidal board.

0, 0, 0, 48, 600, 3108, 10388, 27328, 61668, 124900, 233288, 409008, 681408, 1088388, 1677900, 2509568, 3656428, 5206788, 7266208, 9959600, 13433448, 17858148, 23430468, 30376128, 38952500, 49451428, 62202168, 77574448, 95981648, 117884100

Offset: 1

Author

Vaclav Kotesovec, Jan 07 2011

Keywords

• nonn
• 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 (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A061996, A179403.

Programs

Mathematica

CoefficientList[Series[- 4 x^3 (12 x^6 - 67 x^5 + 140 x^4 - 112 x^3 - 21 x^2 + 66 x + 12) / (x - 1)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 01 2013 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,0,0,48,600,3108,10388,27328,61668,124900},30] (* Harvey P. Dale, Aug 04 2024 *)

Formula

Explicit formula: a(n) = 1/6*n^2*(n^4 -27*n^2 +194), n>=4.

G.f.: -4*x^4*(12*x^6 -67*x^5 +140*x^4 -112*x^3 -21*x^2 +66*x +12)/(x-1)^7.