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A094555 Number of walks of length n between two vertices on the same triangular face of a truncated tetrahedron (triangular prism).

%I A094555 #23 Dec 26 2021 02:59:52
%S A094555 0,1,1,6,11,46,111,386,1051,3366,9671,29866,87891,267086,794431,
%T A094555 2396946,7163531,21545206,64526391,193797626,580955971,1743741726,
%U A094555 5229477551,15691927906,47068793211,141220360646,423633119911,1270955283786
%N A094555 Number of walks of length n between two vertices on the same triangular face of a truncated tetrahedron (triangular prism).
%C A094555 Average of binomial and inverse binomial transforms of the Jacobsthal numbers A001045. - _Paul Barry_, Jan 04 2005
%H A094555 Andrew Howroyd, <a href="/A094555/b094555.txt">Table of n, a(n) for n = 0..1000</a>
%H A094555 R. J. Mathar, <a href="/A102518/a102518.pdf">Counting Walks on Finite Graphs</a>, Nov 2020, Section 3.
%H A094555 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,5,-6).
%F A094555 G.f.: x*(1 - x - x^2)/((1 - x)*(1 + 2*x)*(1 - 3*x)).
%F A094555 a(n) = 3^n/6 - (-2)^n/6 + 1/6 - 0^n/6.
%F A094555 a(n) = 2*a(n-1) + 5*a(n-2) - 6*a(n-3) for n >= 4.
%F A094555 a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*A001045(n-2k). - _Paul Barry_, Jan 04 2005
%F A094555 E.g.f.: exp(-2*x)*(exp(5*x) + exp(3*x) - exp(2*x) - 1)/6. - _Stefano Spezia_, Dec 26 2021
%t A094555 LinearRecurrence[{2, 5, -6}, {0, 1, 1, 6}, 30] (* _Greg Dresden_, Jun 19 2021 *)
%o A094555 (PARI) a(n) = if(n==0, 0, (3^n - (-2)^n + 1)/6) \\ _Andrew Howroyd_, Jun 15 2021
%Y A094555 Cf. A001045, A094554, A094556.
%K A094555 easy,nonn
%O A094555 0,4
%A A094555 _Paul Barry_, May 11 2004