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A102091 Number of perfect matchings in the C_{2n} X P_3 graph (C_{2n} is the cycle graph on 2n vertices and P_3 is the path graph on 3 vertices).

%I A102091 #32 Dec 09 2022 18:21:07
%S A102091 12,32,108,392,1452,5408,20172,75272,280908,1048352,3912492,14601608,
%T A102091 54493932,203374112,759002508,2832635912,10571541132,39453528608,
%U A102091 147242573292,549516764552,2050824484908,7653781175072,28564300215372,106603419686408,397849378530252
%N A102091 Number of perfect matchings in the C_{2n} X P_3 graph (C_{2n} is the cycle graph on 2n vertices and P_3 is the path graph on 3 vertices).
%C A102091 Conjecture: For an integer n > 6, primes of the form either 12k+5 or 12k+7 will never appear in factors of any centered n-gonal number, if and only if n is in A102091. - _Steven Lu_, Nov 15 2022
%H A102091 Colin Barker, <a href="/A102091/b102091.txt">Table of n, a(n) for n = 1..1000</a>
%H A102091 H. Hosoya and A. Motoyama, <a href="https://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 (eq. (53) and Table VII).
%H A102091 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,1).
%F A102091 a(n) = A102089(2*n, 3*n).
%F A102091 a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) with a(1)=12, a(2)=32 and a(3)=108.
%F A102091 G.f.: 4*x*(3-x)*(1-2*x)/((1-x)*(1-4*x+x^2)).
%F A102091 a(n) = A001353(n+1) - 7*A001353(n-1) + 4. - _Ralf Stephan_, May 17 2007
%F A102091 a(n) = 4*A102206(n-1). - _R. J. Mathar_, Mar 06 2016
%F A102091 a(n) = 2*(2 + (2-sqrt(3))^n + (2+sqrt(3))^n). - _Colin Barker_, Nov 03 2016
%F A102091 E.g.f.: -8 + 4*exp(x)*(1 + exp(x)*cosh(sqrt(3)*x)). - _Stefano Spezia_, Oct 27 2019
%e A102091 a(1)=12 because in the graph C_2 X P_3 with vertex set {A,B,C,A',B',C'} and edge set {AB,AC,A'B',B'C',a,a',b,b',c,c'}, where a and a' are two edges between A and A', b and b' are two edges between B and B' and c and c' are two edges between C and C', we have the following twelve perfect matchings: eight matchings by taking one edge from each of the pairs {a,a'},{b,b'} and {c,c'}; two matchings by taking AB, A'B' and either edge from the pair {c,c'}; two matchings by taking BC, B'C' and either edge from the pair {a,a'}.
%p A102091 a[1]:=12: a[2]:=32: a[3]:=108: for n from 4 to 31 do a[n]:=5*a[n-1] -5*a[n-2]+a[n-3] od:seq(a[n],n=1..25);
%t A102091 CoefficientList[Series[4(3-x)(1-2x)/((1-x)(1-4x+x^2)), {x, 0, 25}], x] (* _Vincenzo Librandi_, Nov 04 2016 *)
%t A102091 LinearRecurrence[{5,-5,1}, {12,32,108}, 31] (* _G. C. Greubel_, Oct 27 2019 *)
%o A102091 (PARI) Vec(4*x*(3-x)*(1-2*x)/((1-x)*(1-4*x+x^2)) + O(x^30)) \\ _Colin Barker_, Nov 03 2016
%o A102091 (Magma) [Floor(2*(2+(2-Sqrt(3))^n+(2+Sqrt(3))^n)): n in [1..25]]; // _Vincenzo Librandi_, Nov 04 2016
%o A102091 (Sage)
%o A102091 def A102091_list(prec):
%o A102091     P.<x> = PowerSeriesRing(ZZ, prec)
%o A102091     return P( 4*x*(3-x)*(1-2*x)/((1-x)*(1-4*x+x^2)) ).list()
%o A102091 a=A102091_list(30); a[1:] # _G. C. Greubel_, Oct 27 2019
%o A102091 (GAP) a:=[12,32,108];; for n in [4..30] do a[n]:=5*a[n-1]-5*a[n-2] + a[n-3]; od; a; # _G. C. Greubel_, Oct 27 2019
%Y A102091 Cf. A102089.
%K A102091 nonn,easy
%O A102091 1,1
%A A102091 _Emeric Deutsch_, Dec 29 2004