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A252054 Number of perfect matchings in the P_4 X C_n graph. 6

%I #14 Jan 10 2018 16:00:19

%S 19,121,176,725,1471,5041,11989,37584,97021,290521,783511,2289869,

%T 6323504,18241441,51026011,146160725,411720121,1174844176,3322046089,

%U 9459791909,26804466571,76241702161,216275875376,614789884829,1745053751719

%N Number of perfect matchings in the P_4 X C_n graph.

%H Colin Barker, <a href="/A252054/b252054.txt">Table of n, a(n) for n = 3..1000</a>

%H H. Hosoya and A. Motoyama, <a href="http://dx.doi.org/10.1063/1.526778">An effective algorithm for obtaining polynomials for dimer statistics. Application of operator technique on the topological index to two- and three-dimensional rectangular and torus lattices</a>, J. Math. Physics 26 (1985) 157-167 (Table V).

%H H. Narumi, H. Hosoya, H. Murakami, <a href="http://dx.doi.org/10.1063/1.529254">Generalized expression for the numbers of perfect matching of cylindrical m x n graphs</a>, J. Math. Physics, 32 (1991), 1885-1889.

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (1,13,-7,-61,12,128,0,-128,-12,61,7,-13,-1,1).

%F a(n) = product(13-14*cos(2*(2*j-1)*Pi/n)+2*cos(4*(2*j-1)*Pi/n), j=1..floor(n/2)).

%F a(n) = a(n-1)+13*a(n-2)-7*a(n-3)-61*a(n-4)+12*a(n-5)+128*a(n-6)-128*a(n-8) -12*a(n-9)+61*a(n-10)+7*a(n-11)-13*a(n-12)-a(n-13)+a(n-14).

%F G.f.: -x^3*(29*x^13 -28*x^12 -362*x^11 +175*x^10 +1596*x^9 -198*x^8 -3016*x^7 -248*x^6 +2530*x^5 +464*x^4 -891*x^3 -192*x^2 +102*x +19) / ((x -1)*(x +1)*(x^4 -x^3 -5*x^2 -x +1)*(x^4 -x^3 -3*x^2 +x +1)*(x^4 +x^3 -3*x^2 -x +1)). - _Colin Barker_, Dec 13 2014

%F a(n) ~ ((1 + sqrt(29) + sqrt(14+2*sqrt(29)))/4)^n. - _Vaclav Kotesovec_, Dec 13 2014

%o (PARI) Vec(-x^3*(29*x^13 -28*x^12 -362*x^11 +175*x^10 +1596*x^9 -198*x^8 -3016*x^7 -248*x^6 +2530*x^5 +464*x^4 -891*x^3 -192*x^2 +102*x +19) / ((x -1)*(x +1)*(x^4 -x^3 -5*x^2 -x +1)*(x^4 -x^3 -3*x^2 +x +1)*(x^4 +x^3 -3*x^2 -x +1)) + O(x^100)) \\ _Colin Barker_, Dec 13 2014

%Y Cf. A068397, A102091.

%K nonn,easy

%O 3,1

%A _Sergey Perepechko_, Dec 13 2014

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Last modified March 28 16:58 EDT 2024. Contains 371254 sequences. (Running on oeis4.)