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A384648 Expansion of (1+2*x+x^2) / (1-x-5*x^2-2*x^3).

%I A384648 #7 Jul 08 2025 11:57:59
%S A384648 1,3,9,26,77,225,662,1941,5701,16730,49117,144169,423214,1242293,
%T A384648 3646701,10704594,31422685,92239057,270761670,794802325,2333088789,
%U A384648 6848623754,20103672349,59012968697,173228577950,508500766133,1492669593277,4381630579842
%N A384648 Expansion of (1+2*x+x^2) / (1-x-5*x^2-2*x^3).
%C A384648 Number of walks of length n starting at vertex 0 in the following graph:
%C A384648     1---2
%C A384648    /|\  |
%C A384648   0 | \ |
%C A384648    \|  \|
%C A384648     4---3.
%C A384648 Also, by symmetry, the number of walks of length n starting at vertex 4 in the same graph.
%H A384648 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384648 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,2).
%e A384648 a(2)=9 because we have the walks 3-1-0, 3-1-2, 3-1-3, 3-1-4, 3-2-1, 3-2-3, 3-4-0, 3-4-1, 3-4-3.
%p A384648 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>>)[4,1]:
%p A384648 seq(a(n), n=0..32);
%t A384648 CoefficientList[Series[(1+2*x+x^2) / (1-x-5*x^2-2*x^3), {x, 0, 32}], x]
%Y A384648 Cf. A384646 (vertices 0 and 2), A384647 (vertex 1), A077937 (missing edge {1,3}).
%K A384648 nonn,easy,walk
%O A384648 0,2
%A A384648 _Sean A. Irvine_, Jun 05 2025