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A094554 Number of closed walks of length n at a base vertex of a truncated tetrahedron (triangular prism).

%I A094554 #34 Aug 04 2024 03:13:44
%S A094554 1,0,3,2,19,30,143,322,1179,3110,10183,28842,89939,262990,802623,
%T A094554 2380562,7196299,21479670,64657463,193535482,581480259,1742693150,
%U A094554 5231574703,15687733602,47077181819,141203583430,423666674343
%N A094554 Number of closed walks of length n at a base vertex of a truncated tetrahedron (triangular prism).
%C A094554 For n > 0, 6*a(n) is the number of 3-colorings of the prism of size 2 X n (i.e., C_2 X C_n).More generally, the number of k-colorings of the prism of size 2 X n is given by (k^2 - 3*k + 3)^n + (k - 1) * ((3 - k)^n + (1 - k)^n) + k^2 - 3*k + 1 (chromatic polynomial). - _Sela Fried_, Oct 07 2023
%H A094554 Andrew Howroyd, <a href="/A094554/b094554.txt">Table of n, a(n) for n = 0..1000</a>
%H A094554 N. L. Biggs, R. M. Damerell, and D. A. Sands, <a href="https://doi.org/10.1016/0095-8956(72)90016-0">Recursive families of graphs</a>, Journal of Combinatorial Theory Series B Volume 12 (1972), 123-131.
%H A094554 R. J. Mathar, <a href="/A102518/a102518.pdf">Counting Walks on Finite Graphs</a>, Nov 2020, Section 3.
%H A094554 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,5,-6).
%F A094554 G.f.: (1 - 2*x - 2*x^2 + 2*x^3)/((1 - x)*(1 + 2*x)*(1 - 3*x)).
%F A094554 a(n) = 1/6 + 3^n/6 + (-2)^n/3 for n > 0.
%F A094554 a(n) = 2*a(n-1) + 5*a(n-2) - 6*a(n-3) for n >= 4.
%F A094554 E.g.f.: exp(-2*x)*(1 + exp(2*x))*(2 + exp(3*x))/6. - _Stefano Spezia_, Sep 26 2023
%t A094554 LinearRecurrence[{2, 5, -6}, {1, 0, 3, 2}, 30] (* _Greg Dresden_, Jun 19 2021 *)
%o A094554 (PARI) a(n) = if(n==0, 1, (1 + 3^n + 2*(-2)^n)/6) \\ _Andrew Howroyd_, Jun 14 2021
%Y A094554 Cf. A002001, A025192, A094555, A094556, A328778.
%K A094554 easy,nonn
%O A094554 0,3
%A A094554 _Paul Barry_, May 11 2004