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%I #27 Sep 08 2022 08:45:45
%S 0,3,42,420,3720,31248,256032,2072640,16679040,133824768,1072169472,
%T 8583644160,68694312960,549655154688,4397643866112,35182761492480,
%U 281468534292480,2251774043947008,18014295430397952,144114775759257600
%N Number of isomorphism classes of connected (D_4)-fold coverings of a connected graph with circuit rank n.
%C Previous name: "Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition."
%C From _Álvar Ibeas_, Oct 30 2015: (Start)
%C Also, number of isomorphism classes of connected (C_4 x C_2)-fold coverings of a connected graph with circuit rank n.
%C Also, number of lattices L in Z^n such that the quotient group Z^n / L is C_4 x C_2.
%C Also, number of subgroups of (C_4)^n isomorphic to C_4 x C_2.
%C (End)
%D J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.
%H Álvar Ibeas, <a href="/A160873/b160873.txt">Table of n, a(n) for n = 1..1000</a>
%H J. H. Kwak, J.-H. Chun, and J. Lee, <a href="http://dx.doi.org/10.1137/S0895480196304428">Enumeration of regular graph coverings having finite abelian covering transformation groups</a>, SIAM J. Discrete Math. 11(2), 1998, pp. 273-285.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (14,-56,64).
%F a(n) = 2^(n-2) * (2^n - 1) * (2^(n-1) - 1) = 8^(n-1) - 6*4^(n-2) + 2^(n-2) [Kwak, Chun, and Lee]. - _Álvar Ibeas_, Oct 30 2015
%F From _Colin Barker_, Oct 30 2015: (Start)
%F a(n) = 2^(-3+n) * (2-3*2^n+4^n).
%F a(n) = 14*a(n-1)-56*a(n-2)+64*a(n-3) for n>3.
%F G.f.: -3*x^2/((2*x-1)*(4*x-1)*(8*x-1)). (End)
%t Table[2^(-3+n)*(2-3*2^n+4^n), {n,1,30}] (* or *) LinearRecurrence[{14, -56, 64}, {0, 3, 42}, 30] (* _G. C. Greubel_, Apr 30 2018 *)
%o (PARI) a(n) = 2^(-3+n) * (2-3*2^n+4^n) \\ _Colin Barker_, Oct 30 2015
%o (PARI) concat(0, Vec(-3*x^2/((2*x-1)*(4*x-1)*(8*x-1)) + O(x^30))) \\ _Colin Barker_, Oct 30 2015
%o (Magma) [2^(-3+n)*(2-3*2^n+4^n): n in [1..30]]; // _G. C. Greubel_, Apr 30 2018
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_, Nov 15 2009
%E Name clarified by _Álvar Ibeas_, Oct 30 2015
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