|
%I #56 Oct 01 2021 18:04:37
%S 0,0,8,44,140,340,700,1288,2184,3480,5280,7700,10868,14924,20020,
%T 26320,34000,43248,54264,67260,82460,100100,120428,143704,170200,
%U 200200,234000,271908,314244,361340,413540,471200,534688,604384
%N Number of ways to place two nonattacking queens on an n X n board.
%H Vincenzo Librandi, <a href="/A036464/b036464.txt">Table of n, a(n) for n = 1..1000</a>
%H S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, <a href="http://www.math.binghamton.edu/zaslav/Tpapers/qq1.pdf">A q-queens problem I. General theory</a>, January 26, 2013. - _N. J. A. Sloane_, Feb 16 2013
%H S. Chaiken, C. R. H. Hanusa and T. Zaslavsky, <a href="http://arxiv.org/abs/1609.00853">A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks)</a>, arXiv:1609.00853 [math.CO], Sep 03 2016.
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Non-attacking chess pieces</a>
%H I. Rivin, I. Vardi and P. Zimmermann, <a href="http://www.jstor.org/stable/2974691">The n-queens problem</a>, Amer. Math. Monthly, 101 (1994), 629-639.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = C(n, 3)*(3*n-1).
%F G.f.: 4*x^3*(2+x)/(1-x)^5. - _Colin Barker_, May 02 2012
%F a(n) = 2*sum_{i=1..n-2} i(i + 1)^2. - _Wesley Ivan Hurt_, Mar 18 2014
%F E.g.f.: (exp(x) * x^3 * (8 + 3*x))/6. - _Vaclav Kotesovec_, Feb 15 2015
%F For n>0, a(n) = A163102(n-1) - A006331(n-1). - _Antal Pinter_, Sep 20 2015
%p 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
%t f[k_] := 2 k; t[n_] := Table[f[k], {k, 1, n}]
%t a[n_] := SymmetricPolynomial[2, t[n]]
%t Table[a[n], {n, 2, 50}] (* A036464 *)
%t Table[a[n]/4, {n, 2, 50}] (* A000914 *)
%t (* _Clark Kimberling_, Dec 31 2011 *)
%t CoefficientList[Series[4 x^2 (2 + x) / (1-x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, May 02 2013 *)
%t LinearRecurrence[{5,-10,10,-5,1},{0,0,8,44,140},50] (* _Harvey P. Dale_, Mar 26 2015 *)
%Y Cf. A047659, A061994, A108792, A176186, A178721.
%Y Column k=2 of A348129.
%K nonn,easy,nice
%O 1,3
%A _Robert G. Wilson v_, Raymond Bush (c17h21no4(AT)hotmail.com), Kirk Conely, _N. J. A. Sloane_
|
