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
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7).
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
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
Geoffrey Critzer, Oct 27 2019
STATUS
approved