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

Offset: 1

Author

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

Keywords

• nonn
• easy
• nice

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, A q-queens problem I. General theory, January 26, 2013. - _N. J. A. Sloane_, Feb 16 2013
• S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks), arXiv:1609.00853 [math.CO], Sep 03 2016.
• V. Kotesovec, Non-attacking chess pieces
• I. Rivin, I. Vardi and P. Zimmermann, The n-queens problem, Amer. Math. Monthly, 101 (1994), 629-639.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

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

Column k=2 of A348129.

Programs

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 *)

Formula

a(n) = C(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