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Number of ways to place 2 nonattacking nightriders on an n X n board.
8

%I #18 Apr 23 2022 18:37:23

%S 0,6,28,96,240,518,980,1712,2784,4310,6380,9136,12688,17206,22820,

%T 29728,38080,48102,59964,73920,90160,108966,130548,155216,183200,

%U 214838,250380,290192,334544,383830,438340,498496,564608,637126

%N Number of ways to place 2 nonattacking nightriders on an n X n board.

%C A nightrider is a fairy chess piece that can move (proportionate to how a knight moves) in any direction.

%D Christian Poisson, Echecs et mathematiques, Rex Multiplex 29/1990, p.829

%H Vincenzo Librandi, <a href="/A172141/b172141.txt">Table of n, a(n) for n = 1..1000</a>

%H Christopher R. H. Hanusa, T. Zaslavsky, and S. Chaiken, <a href="http://arxiv.org/abs/1609.00853">A q-Queens Problem. IV. Queens, Bishops, Nightriders (and Rooks)</a>, arXiv preprint arXiv:1609.00853, a12016

%H Vaclav Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Number of ways of placing non-attacking queens and kings on boards of various sizes</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).

%F Explicit formula (Christian Poisson, 1990): a(n) = n*(3*n^3 - 5*n^2 + 9*n - 4)/6 if n is even and a(n) = n*(n - 1)*(3*n^2 - 2*n + 7)/6 if n is odd.

%F G.f.: 2*x^2*(3+2*x+x^2)*(1+x+2*x^2)/((1-x)^5*(1+x)^2). - _Vaclav Kotesovec_, Mar 25 2010

%F From _G. C. Greubel_, Apr 21 2022: (Start)

%F a(n) = (1/12)*n*(3*(-1)^n - (11 - 18*n + 10*n^2 - 6*n^3)).

%F E.g.f.: (x/12)*(-3*exp(-x) + (3 + 30*x + 26*x^2 + 6*x^3)exp(x)). (End)

%t CoefficientList[Series[2*x*(3+2*x+x^2)*(1+x+2*x^2)/((1-x)^5*(1+x)^2), {x,0,40}], x] (* _Vincenzo Librandi_, May 27 2013 *)

%o (Magma) [(n/12)*(3*(-1)^n -(11 -18*n +10*n^2 -6*n^3)): n in [1..40]]; // _G. C. Greubel_, Apr 21 2022

%o (SageMath) [(n/12)*(3*(-1)^n -(11 -18*n +10*n^2 -6*n^3)) for n in (1..40)] # _G. C. Greubel_, Apr 21 2022

%Y Cf. A036464, A172123, A172132, A172137.

%K easy,nonn

%O 1,2

%A _Vaclav Kotesovec_, Jan 26 2010