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Number of perfect matchings in the graph C_5 X C_{2n}.
11

%I #23 Feb 14 2021 05:51:44

%S 722,9922,155682,2540032,41934482,694861522,11527389122,191304901282,

%T 3175220160032,52703408458882,874800747092322,14520494659638322,

%U 241020661471736882,4000620276282860032,66404949893677073282,1102233473331064193122,18295603728585969257522

%N Number of perfect matchings in the graph C_5 X C_{2n}.

%H Colin Barker, <a href="/A231485/b231485.txt">Table of n, a(n) for n = 2..819</a>

%H P. W. Kasteleyn, <a href="http://dx.doi.org/10.1016/0031-8914(61)90063-5">The Statistics of Dimers on a Lattice</a>, Physica, 27 (1961), 1209-1225.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (29,-261,1029,-2001,2001,-1029,261,-29,1).

%F G.f.: 2*x^2*(361-5508*x+28193*x^2-64021*x^3+70770*x^4-38841*x^5+10278*x^6-1173*x^7+41*x^8)/((1-x)*(1-9*x+21*x^2-9*x^3+x^4)*(1-19*x+41*x^2-19*x^3+x^4)).

%F From _Seiichi Manyama_, Feb 14 2021: (Start)

%F a(n) = sqrt( Product_{j=1..n} Product_{k=1..5} (4*sin((2*j-1)*Pi/(2*n))^2 + 4*sin((2*k-1)*Pi/5)^2) ).

%F a(n) = 28*a(n-1) - 233*a(n-2) + 796*a(n-3) - 1205*a(n-4) + 796*a(n-5) - 233*a(n-6) + 28*a(n-7) - a(n-8) + 200. (End)

%o (PARI) Vec(2*x^2*(361-5508*x+28193*x^2-64021*x^3+70770*x^4-38841*x^5+10278*x^6-1173*x^7+41*x^8)/((1-x)*(1-9*x+21*x^2-9*x^3+x^4)*(1-19*x+41*x^2-19*x^3+x^4)) + O(x^100)) \\ _Colin Barker_, Dec 13 2014

%o (PARI) default(realprecision, 120);

%o a(n) = round(sqrt(prod(j=1, n, prod(k=1, 5, 4*sin((2*j-1)*Pi/(2*n))^2+4*sin((2*k-1)*Pi/5)^2)))); \\ _Seiichi Manyama_, Feb 14 2021

%Y Cf. A220864, A231087.

%K nonn,easy

%O 2,1

%A _Sergey Perepechko_, Nov 09 2013