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A194646 Number of ways to place 4n nonattacking kings on an 8 X 2n cylindrical chessboard.

%I A194646 #26 Aug 17 2024 15:10:02
%S A194646 80,276,1082,4460,18890,81606,358564,1599820,7238864,33175486,
%T A194646 153802520,720390254,3404944506,16221905696,77820675992,375564803020,
%U A194646 1821845982082,8876847931644,43416046650306,213033152875350,1048198981050148,5169676077206180
%N A194646 Number of ways to place 4n nonattacking kings on an 8 X 2n cylindrical chessboard.
%C A194646 This cylinder is horizontal: a chessboard where it is supposed that rows 1 and 2n are in contact (number of columns = 8, number of rows = 2n).
%H A194646 Ray Chandler, <a href="/A194646/b194646.txt">Table of n, a(n) for n = 1..1430</a>
%H A194646 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 A194646 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (29, -369, 2708, -12676, 39539, -83449, 118727, -111545, 66422, -23320, 4235, -300).
%F A194646 a(n) = 4 + 2*5^n + 2*4^n + 2*(2+sqrt(3))^n+2*(2-sqrt(3))^n + 4*((5+sqrt(5))/2)^n+4*((5-sqrt(5))/2)^n + 4*((5+sqrt(13))/2)^n+4*((5-sqrt(13))/2)^n + 2*(2*cos(Pi/7))^(2n)+2*(2*cos(2*Pi/7))^(2n)+2*(2*cos(3*Pi/7))^(2n).
%F A194646 Recurrence: a(n) = -300*a(n-12) + 4235*a(n-11) - 23320*a(n-10) + 66422*a(n-9) - 111545*a(n-8) + 118727*a(n-7) - 83449*a(n-6) + 39539*a(n-5) - 12676*a(n-4) + 2708*a(n-3) - 369*a(n-2) + 29*a(n-1).
%F A194646 G.f.: -2*(-17 + 453*x - 5251*x^2 + 34737*x^3 - 144635*x^4 + 394423*x^5 - 711101*x^6 + 836705*x^7 - 620007*x^8 + 270365*x^9 - 61055*x^10 + 5335*x^11)/((-1+x)*(-1+4*x)*(-1+5*x)*(1-4*x+x^2)*(1-5*x+3*x^2)*(1-5*x+5*x^2)*(-1+5*x-6*x^2+x^3)).
%t A194646 Table[FullSimplify[4+2*5^n+2*4^n + 2*(2+Sqrt[3])^n + 2*(2-Sqrt[3])^n+ 4*((5+Sqrt[5])/2)^n + 4*((5-Sqrt[5])/2)^n + 4*((5+Sqrt[13])/2)^n+4*((5-Sqrt[13])/2)^n+ 2*(2*Cos[Pi/7])^(2n) + 2*(2*Cos[2*Pi/7])^(2n) + 2*(2*Cos[3*Pi/7])^(2n)], {n,10}]
%Y A194646 Cf. A173782, A194644, A194645, A137432, A195592.
%K A194646 nonn
%O A194646 1,1
%A A194646 _Vaclav Kotesovec_, Aug 31 2011