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A384647 Expansion of (1+3*x+x^2) / (1-x-5*x^2-2*x^3).

%I A384647 #6 Jul 08 2025 11:57:04
%S A384647 1,4,10,32,90,270,784,2314,6774,19912,58410,171518,503392,1477802,
%T A384647 4337798,12733592,37378186,109721742,322079856,945444938,2775287702,
%U A384647 8146672104,23914000490,70197936414,206061283072,604878966122,1775581254310,5212098651064
%N A384647 Expansion of (1+3*x+x^2) / (1-x-5*x^2-2*x^3).
%C A384647 Number of walks of length n starting at vertex 1 in the following graph:
%C A384647     1---2
%C A384647    /|\  |
%C A384647   0 | \ |
%C A384647    \|  \|
%C A384647     4---3.
%H A384647 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384647 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,2).
%e A384647 a(2)=10 because we have the walks 1-0-1, 1-0-4, 1-2-1, 1-2-3, 1-3-1, 1-3-2, 1-3-4, 1-4-0, 1-4-1, 1-4-3.
%p A384647 a:= n->  (<<0|1|0|0|1>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <1|1|0|1|0>>^n. <<1,1,1,1,1>>)[2,1]:
%p A384647 seq(a(n), n=0..32);
%t A384647 CoefficientList[Series[(1+3*x+x^2) / (1-x-5*x^2-2*x^3), {x, 0, 32}], x]
%Y A384647 Cf. A384646 (vertices 0, 2), A384648 (vertices 3 and 4), A077937 (missing edge {1,3}).
%K A384647 nonn,easy,walk
%O A384647 0,2
%A A384647 _Sean A. Irvine_, Jun 05 2025