A061996 Number of ways to place 3 nonattacking kings on an n X n board.

0, 0, 0, 8, 140, 964, 3920, 11860, 29708, 65240, 129984, 240240, 418220, 693308, 1103440, 1696604, 2532460, 3684080, 5239808, 7305240, 10005324, 13486580, 17919440, 23500708, 30456140, 39043144, 49553600, 62316800

Offset: 0

Author

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 31 2001

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Placing non-attacking queens and kings on boards of various sizes, part of "Between chessboard and computer", 1996, pp. 204 - 206.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A061995, A061997, A061998, A193580.

Programs

Mathematica

CoefficientList[Series[4x^3(2 +21x +38x^2 -42x^3 +11x^4)/(1-x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, May 02 2013 *)

SageMath

[(n-1)*(n-2)*(n^4+3*n^3-20*n^2-30*n+132)/6 -44*bool(n==0) for n in (0..40)] # G. C. Greubel, Apr 29 2022

Formula

G.f.: 4*x^3*(2 + 21*x + 38*x^2 - 42*x^3 + 11*x^4)/(1 - x)^7.

Recurrence: a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7), n >= 8.

a(n) = (n-1)*(n-2)*(n^4 + 3*n^3 - 20*n^2 - 30*n + 132)/6, n >= 1.

a(n) = A193580(n,3). - R. J. Mathar, Sep 03 2016

E.g.f.: -44 + (1/6)*(264 -264*x +132*x^2 -36*x^3 +38*x^4 +15*x^5 +x^6)*exp(x). - G. C. Greubel, Apr 29 2022