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

Author

N. J. A. Sloane, Nov 15 2009

Keywords

• nonn
• easy

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).

Programs

Magma

[2^(-3+n)*(2-3*2^n+4^n): n in [1..30]]; // G. C. Greubel, Apr 30 2018

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 *)

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

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)

Extensions

Name clarified by Álvar Ibeas, Oct 30 2015