|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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 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
|
|
|
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(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
|
|
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
|
|
|
|