A160873 Number of isomorphism classes of connected (D_4)-fold coverings of a connected graph with circuit rank n.

0, 3, 42, 420, 3720, 31248, 256032, 2072640, 16679040, 133824768, 1072169472, 8583644160, 68694312960, 549655154688, 4397643866112, 35182761492480, 281468534292480, 2251774043947008, 18014295430397952, 144114775759257600

Offset

1,2

Comments

Previous name: "Regular coverings having dihedral voltage groups: see Kwak-Lee reference in A160870 for precise definition."

From Álvar Ibeas, Oct 30 2015: (Start)

Also, number of isomorphism classes of connected (C_4 x C_2)-fold coverings of a connected graph with circuit rank n.

Also, number of lattices L in Z^n such that the quotient group Z^n / L is C_4 x C_2.

Also, number of subgroups of (C_4)^n isomorphic to C_4 x C_2.

(End)

References

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.

Links

Álvar Ibeas, Table of n, a(n) for n = 1..1000

J. H. Kwak, J.-H. Chun, and J. Lee, Enumeration of regular graph coverings having finite abelian covering transformation groups, SIAM J. Discrete Math. 11(2), 1998, pp. 273-285.

Index entries for linear recurrences with constant coefficients, signature (14,-56,64).

Formula

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

From Colin Barker, Oct 30 2015: (Start)

a(n) = 2^(-3+n) * (2-3*2^n+4^n).

a(n) = 14*a(n-1)-56*a(n-2)+64*a(n-3) for n>3.

G.f.: -3*x^2/((2*x-1)*(4*x-1)*(8*x-1)). (End)

Mathematica

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 *)

Prog

(PARI) a(n) = 2^(-3+n) * (2-3*2^n+4^n) \\ Colin Barker, Oct 30 2015
(PARI) concat(0, Vec(-3*x^2/((2*x-1)*(4*x-1)*(8*x-1)) + O(x^30))) \\ Colin Barker, Oct 30 2015
(Magma) [2^(-3+n)*(2-3*2^n+4^n): n in [1..30]]; // G. C. Greubel, Apr 30 2018

Crossrefs

Sequence in context: A119577 A051273 A350901 * A084512 A084522 A160874

Adjacent sequences: A160870 A160871 A160872 * A160874 A160875 A160876

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 15 2009

Extensions

Name clarified by Álvar Ibeas, Oct 30 2015

Status

approved