A061990 Number of ways to place 4 nonattacking queens on a 4 X n board.

0, 0, 0, 0, 2, 12, 46, 140, 344, 732, 1400, 2468, 4080, 6404, 9632, 13980, 19688, 27020, 36264, 47732, 61760, 78708, 98960, 122924, 151032, 183740, 221528, 264900, 314384, 370532, 433920, 505148, 584840, 673644, 772232, 881300, 1001568, 1133780, 1278704, 1437132

Offset: 0

Author

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), May 29 2001

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• V. Kotesovec, Ways of placing non-attacking queens and kings..., part of "Between chessboard and computer", 1996, pp. 204 - 206.
• E. Lucas, Recreations mathematiques I, Albert Blanchard, Paris, 1992, p. 231.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A061989.

Programs

Mathematica

Join[{0,0,0,0,2,12,46},LinearRecurrence[{5,-10,10,-5,1},{140,344,732,1400,2468},30]] (* Harvey P. Dale, Mar 06 2013 *)
CoefficientList[Series[-2 x^4 (x^3 - x^2 + x + 1) (x^4 + 4 x^2 + 1) / (x-1)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 02 2013 *)

PARI

a(n)=if(n<7,[0, 0, 0, 0, 2, 12, 46][n+1],n^4-18*n^3+139*n^2-534*n+840) \\ Charles R Greathouse IV, Oct 21 2022

Formula

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

Recurrence: a(n)=5*a(n-1)-10*a(n-2)+10*a(n-3)-5*a(n-4)+a(n-5), n >= 12.

Explicit formula (H. Tarry, 1890): a(n)=n^4-18*n^3+139*n^2-534*n+840, n >= 7.