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%I #25 Feb 13 2020 06:46:43
%S 12,32,90,256,732,2102,6060,17536,50922,148352,433500,1270246,3731532,
%T 10987232,32418810,95835136,283784412,841611542,2499330540,7431221056,
%U 22118855562,65898914432,196498594140,586358987206,1750864725132,5231094261152,15636995277210
%N Number of ways to place 2n nonattacking kings on a 4 X 2n cylindrical chessboard.
%C This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 4, number of rows = 2n).
%H Indranil Ghosh, <a href="/A194644/b194644.txt">Table of n, a(n) for n = 1..2091</a>
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Number of ways of placing non-attacking queens, kings, bishops and knights</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-10,3).
%F a(n) = 2*3^n + 2*((3+sqrt(5))/2)^n + 2*((3-sqrt(5))/2)^n.
%F Recurrence: a(n) = 3*a(n-3) - 10*a(n-2) + 6*a(n-1).
%F G.f.: -2*(3-12*x+10*x^2)/((-1+3*x)*(1-3*x+x^2)).
%t Table[2*3^n+2*LucasL[2n], {n,25}]
%t Drop[CoefficientList[Series[-2*(3 - 12*x + 10*x^2)/((-1 + 3*x)*(1 - 3*x + x^2)), {x, 0, 27}],x], 1] (* or *) LinearRecurrence[{6,-10,3},{12,32,90},27] (* _Indranil Ghosh_, Mar 05 2017 *)
%o (PARI) print(Vec(-2*(3 - 12*x + 10*x^2)/((-1 + 3*x)*(1 - 3*x + x^2)) + O(x^27))); \\ _Indranil Ghosh_, Mar 05 2017
%Y Cf. A061593, A137432, A195590.
%K nonn,easy
%O 1,1
%A _Vaclav Kotesovec_, Aug 31 2011
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