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%I #18 Sep 26 2023 13:40:17
%S 0,1,0,7,8,51,100,407,1008,3451,9500,30207,87208,268451,791700,
%T 2402407,7152608,21567051,64482700,193885007,580781208,1744091251,
%U 5228778500,15693326007,47065997008,141225953051,423621935100,1270977653407
%N Number of walks of length n between opposite vertices on a triangular prism.
%H Andrew Howroyd, <a href="/A094556/b094556.txt">Table of n, a(n) for n = 0..1000</a>
%H R. J. Mathar, <a href="/A102518/a102518.pdf">Counting Walks on Finite Graphs</a>, Nov 2020, Section 3.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,5,-6).
%F G.f.: x*(1 - 2*x + 2*x^2)/((1 - x)*(1 + 2*x)*(1 - 3*x)).
%F a(n) = 3^n/6 - (-2)^n/3 - 1/6 + 0^n/3.
%F a(n) = 2*a(n-1) + 5*a(n-2) - 6*a(n-3) for n >= 4.
%F E.g.f.: exp(-x)*(2+exp(3*x))*sinh(x)/3. - _Stefano Spezia_, Sep 26 2023
%t LinearRecurrence[{2,5,-6},{0,1,0,7},30] (* or *) CoefficientList[ Series[ x (1-2x+2x^2)/((1-x)(1+2x)(1-3x)),{x,0,30}],x] (* _Harvey P. Dale_, Jul 13 2011 *)
%o (PARI) a(n) = if(n==0, 0, (3^n - 2*(-2)^n - 1)/6) \\ _Andrew Howroyd_, Jun 15 2021
%Y Cf. A094554, A094555.
%K easy,nonn
%O 0,4
%A _Paul Barry_, May 11 2004
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