

A328778


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


4



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

0,2


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 4colorings of the grid graph P_2 X P_(n1). More generally, for q > 1, the number of qcolorings 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



FORMULA

G.f.: 2  1/f(x) where f(x) is the g.f. for A054879.
G.f.: (1  4*x  9*x^2) / (1  7*x).
a(n) = 7*a(n1) 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


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 &]


PROG

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


CROSSREFS



KEYWORD

nonn,walk,easy


AUTHOR



STATUS

approved



