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A212797 Number of spanning trees in C_2 X C_n.

%I A212797 #50 Oct 30 2023 10:39:10
%S A212797 2,32,294,2304,16810,117600,799694,5326848,34928082,226195360,
%T A212797 1450199542,9220780800,58221203066,365440965344,2282085037470,
%U A212797 14187697422336,87860208024994,542209573735200,3335797263902918,20465738163774720,125247216613782858
%N A212797 Number of spanning trees in C_2 X C_n.
%C A212797 From _Harry Richman_ and _Alois P. Heinz_, Jan 31 2023: (Start)
%C A212797 Row 2 of array in A212796.
%C A212797 a(n) is a divisibility sequence, i.e., if m divides n then a(m) divides a(n), since A001108 is one. (End)
%H A212797 Germain Kreweras, <a href="http://dx.doi.org/10.1016/0095-8956(78)90021-7">Complexité et circuits Eulériens dans les sommes tensorielles de graphes</a>, J. Combin. Theory, B 24 (1978), 202-212. See p. 210, Parag. 4.
%H A212797 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (14,-63,100,-63,14,-1).
%F A212797 G.f.: 2*x*(1+2*x-14*x^2+2*x^3+x^4)/((x-1)^2*(1-6*x+x^2)^2) . - _R. J. Mathar_, Apr 16 2018
%F A212797 Conjecture: a(n) = 16*A001109(n+1) +3*A001109(n) -16*(A144133(n)-6*A144133(n-1)) -n = 3*A144133(n-1) -2*A144133(n-2) +3*A144133(n-3) -n. - _R. J. Mathar_, Apr 16 2018
%F A212797 From _Seiichi Manyama_, Jan 13 2021: (Start)
%F A212797 a(n) = 2 * n * A001108(n).
%F A212797 a(n) = 14*a(n-1) - 63*a(n-2) + 100*a(n-3) - 63*a(n-4) + 14*a(n-5) - a(n-6) for n > 6. (End)
%o A212797 (PARI) default(realprecision, 120);
%o A212797 {a(n) = round(n*2^(2*n-1)*prod(k=1, n-1, 1+sin(k*Pi/n)^2))} \\ _Seiichi Manyama_, Jan 13 2021
%o A212797 (PARI) my(N=66, x='x+O('x^N)); Vec(2*x*deriv(x*(1+x)/((1-x)*(1-6*x+x^2)))) \\ _Seiichi Manyama_, Jan 13 2021
%Y A212797 Cf. A001108, A212796.
%K A212797 nonn
%O A212797 1,1
%A A212797 _N. J. A. Sloane_, May 27 2012
%E A212797 More terms from _Seiichi Manyama_, Jan 13 2021