

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).


7



12, 32, 108, 392, 1452, 5408, 20172, 75272, 280908, 1048352, 3912492, 14601608, 54493932, 203374112, 759002508, 2832635912, 10571541132, 39453528608, 147242573292, 549516764552, 2050824484908, 7653781175072, 28564300215372, 106603419686408, 397849378530252
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OFFSET

1,1


COMMENTS

Conjecture: For an integer n > 6, primes of the form either 12k+5 or 12k+7 will never appear in factors of any centered ngonal number, if and only if n is in A102091.  Steven Lu, Nov 15 2022


LINKS



FORMULA

a(n) = 5*a(n1)  5*a(n2) + a(n3) with a(1)=12, a(2)=32 and a(3)=108.
G.f.: 4*x*(3x)*(12*x)/((1x)*(14*x+x^2)).
a(n) = 2*(2 + (2sqrt(3))^n + (2+sqrt(3))^n).  Colin Barker, Nov 03 2016
E.g.f.: 8 + 4*exp(x)*(1 + exp(x)*cosh(sqrt(3)*x)).  Stefano Spezia, Oct 27 2019


EXAMPLE

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'}.


MAPLE

a[1]:=12: a[2]:=32: a[3]:=108: for n from 4 to 31 do a[n]:=5*a[n1] 5*a[n2]+a[n3] od:seq(a[n], n=1..25);


MATHEMATICA

CoefficientList[Series[4(3x)(12x)/((1x)(14x+x^2)), {x, 0, 25}], x] (* Vincenzo Librandi, Nov 04 2016 *)
LinearRecurrence[{5, 5, 1}, {12, 32, 108}, 31] (* G. C. Greubel, Oct 27 2019 *)


PROG

(PARI) Vec(4*x*(3x)*(12*x)/((1x)*(14*x+x^2)) + O(x^30)) \\ Colin Barker, Nov 03 2016
(Magma) [Floor(2*(2+(2Sqrt(3))^n+(2+Sqrt(3))^n)): n in [1..25]]; // Vincenzo Librandi, Nov 04 2016
(Sage)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 4*x*(3x)*(12*x)/((1x)*(14*x+x^2)) ).list()
(GAP) a:=[12, 32, 108];; for n in [4..30] do a[n]:=5*a[n1]5*a[n2] + a[n3]; od; a; # G. C. Greubel, Oct 27 2019


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



