A252054 Number of perfect matchings in the P_4 X C_n graph.
19, 121, 176, 725, 1471, 5041, 11989, 37584, 97021, 290521, 783511, 2289869, 6323504, 18241441, 51026011, 146160725, 411720121, 1174844176, 3322046089, 9459791909, 26804466571, 76241702161, 216275875376, 614789884829, 1745053751719
Offset: 3
Links
- Colin Barker, Table of n, a(n) for n = 3..1000
- H. Hosoya and A. Motoyama, 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, J. Math. Physics 26 (1985) 157-167 (Table V).
- H. Narumi, H. Hosoya, H. Murakami, Generalized expression for the numbers of perfect matching of cylindrical m x n graphs, J. Math. Physics, 32 (1991), 1885-1889.
- Index entries for linear recurrences with constant coefficients, signature (1,13,-7,-61,12,128,0,-128,-12,61,7,-13,-1,1).
Programs
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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
Formula
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)).
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).
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
a(n) ~ ((1 + sqrt(29) + sqrt(14+2*sqrt(29)))/4)^n. - Vaclav Kotesovec, Dec 13 2014