A036464 Number of ways to place two nonattacking queens on an n X n board.

0, 0, 8, 44, 140, 340, 700, 1288, 2184, 3480, 5280, 7700, 10868, 14924, 20020, 26320, 34000, 43248, 54264, 67260, 82460, 100100, 120428, 143704, 170200, 200200, 234000, 271908, 314244, 361340, 413540, 471200, 534688, 604384, 680680, 763980, 854700, 953268, 1060124

Offset

1,3

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Seth Chaiken, Christopher R. H. Hanusa, and Thomas Zaslavsky, A q-Queens Problem. I. General Theory, The Electronic Journal of Combinatorics, Vol. 21, No. 3 (2014), Article P3-33.

Seth Chaiken, Christopher R. H. Hanusa, and Thomas Zaslavsky, A -Queens Problem. V. Some of Our Favorite Pieces: Queens, Bishops, Rooks, and Nightriders, arXiv:1609.00853 [math.CO], 2016-2020.

Vaclav Kotesovec, Non-attacking chess pieces.

Igor Rivin, Ilan Vardi, and Paul Zimmerman, The n-queens problem, Amer. Math. Monthly, Vol. 101, No. 7 (1994), 629-639.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = binomial(n, 3)*(3*n-1).

G.f.: 4*x^3*(2+x)/(1-x)^5. - Colin Barker, May 02 2012

a(n) = 2*Sum_{i=1..n-2} i(i + 1)^2. - Wesley Ivan Hurt, Mar 18 2014

E.g.f.: (exp(x) * x^3 * (8 + 3*x))/6. - Vaclav Kotesovec, Feb 15 2015

For n>0, a(n) = A163102(n-1) - A006331(n-1). - Antal Pinter, Sep 20 2015

From Amiram Eldar, Nov 06 2025: (Start)

Sum_{n>=3} 1/a(n) = 81*log(3)/10 - 9*sqrt(3)*Pi/10 - 96/25.

Sum_{n>=3} (-1)^(n+1)/a(n) = 9*sqrt(3)*Pi/5 - 24*log(2)/5 - 159/25. (End)

Maple

f:=n->n^4/2 - 5*n^3/3 + 3*n^2/2 - n/3; [seq(f(n), n=1..200)]; # N. J. A. Sloane, Feb 16 2013

Mathematica

f[k_] := 2 k; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 50}]   (* A036464 *)
Table[a[n]/4, {n, 2, 50}] (* A000914 *)
(* Clark Kimberling, Dec 31 2011 *)
CoefficientList[Series[4 x^2 (2 + x) / (1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, May 02 2013 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 0, 8, 44, 140}, 50] (* Harvey P. Dale, Mar 26 2015 *)

Crossrefs

Cf. A006331, A047659, A061994, A108792, A163102, A176186, A178721.

Column k=2 of A348129.

Sequence in context: A316466 A100583 A261996 * A000938 A252871 A307044

Adjacent sequences: A036461 A036462 A036463 * A036465 A036466 A036467

Keyword

nonn,easy,nice

Author

Robert G. Wilson v, Raymond Bush (c17h21no4(AT)hotmail.com), Kirk Conely, and N. J. A. Sloane

Status

approved