A328778 Number of indecomposable closed walks of length 2n along the edges of a cube based at a vertex.

1, 3, 12, 84, 588, 4116, 28812, 201684, 1411788, 9882516, 69177612, 484243284, 3389702988, 23727920916, 166095446412, 1162668124884, 8138676874188, 56970738119316, 398795166835212, 2791566167846484, 19540963174925388

Offset: 0

Author

Geoffrey Critzer, Oct 27 2019

Keywords

• nonn
• walk
• easy

Comments

An indecomposable closed walk returns to its starting vertex exactly once (on the final step).

For n > 1, a(n) is the number of 4-colorings of the grid graph P_2 X P_(n-1). More generally, for q > 1, the number of q-colorings of the grid graph P_2 X P_n is given by q*(q - 1)*((q - 1)*(q - 2) + 1)^(n - 1). - Sela Fried, Sep 25 2023

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7).

Crossrefs

Cf. A054879, A025192.

Programs

Mathematica

nn = 40; list = Range[0, nn]! CoefficientList[Series[ Cosh[x]^3, {x, 0, nn}], x]; a = Sum[list[[i]] x^(i - 1), {i, 1, nn + 1}]; Select[CoefficientList[Series[ 2 - 1/a, {x, 0, nn}], x], # > 0 &]

PARI

Vec((1 - 4*x - 9*x^2) / (1 - 7*x) + O(x^25)) \\ Colin Barker, Oct 28 2019

Formula

G.f.: 2 - 1/f(x) where f(x) is the g.f. for A054879.

From Colin Barker, Oct 27 2019: (Start)

G.f.: (1 - 4*x - 9*x^2) / (1 - 7*x).

a(n) = 7*a(n-1) for n>2.

a(n) = 12*7^(n - 2) for n>1.

(End)

E.g.f.: (1/49)*(37 + 12*exp(7*x) + 63*x). - Stefano Spezia, Oct 27 2019