A061989 Number of ways to place 3 nonattacking queens on a 3 X n board.

0, 0, 0, 0, 4, 14, 36, 76, 140, 234, 364, 536, 756, 1030, 1364, 1764, 2236, 2786, 3420, 4144, 4964, 5886, 6916, 8060, 9324, 10714, 12236, 13896, 15700, 17654, 19764, 22036, 24476, 27090, 29884, 32864, 36036, 39406, 42980, 46764, 50764

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
• Vaclav 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 (4,-6,4,-1).

Crossrefs

Cf. A061990, A117937, A229183.

Essentially the same as A079908.

Programs

Magma

[0,0,0] cat [(n-3)*(n^2-6*n+12): n in [3..50]]; // G. C. Greubel, Apr 29 2022

Maple

A061989 := proc(n)
    if n >= 3 then
        (n-3)*(n^2-6*n+12) ;
    else
        0;
    end if;
end proc:
seq(A061989(n),n=0..30) ; # R. J. Mathar, Aug 16 2019

Mathematica

CoefficientList[Series[2*x^4*(2-x+2*x^2)/(1-x)^4, {x, 0, 50}], x] (* Vincenzo Librandi, May 02 2013 *)

SageMath

[0,0,0]+[(n-3)*((n-3)^2 +3) for n in (3..50)] # G. C. Greubel, Apr 29 2022

Formula

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

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n >= 7.

Explicit formula (H. Tarry, 1890): a(n) = (n-3)*(n^2-6*n+12), n >= 3.

(4, 14, 36, ...) is the binomial transform of row 4 of A117937: (4, 10, 12, 6). - Gary W. Adamson, Apr 09 2006

a(n) = 2*A229183(n-3). - R. J. Mathar, Aug 16 2019

E.g.f.: 36 + 14*x + 2*x^2 + (-36 + 22*x - 6*x^2 + x^3)*exp(x). - G. C. Greubel, Apr 29 2022