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A384641 Expansion of (1+2*x-x^3) / (1-x-5*x^2+x^3+2*x^4).

%I A384641 #6 Jul 08 2025 11:54:18
%S A384641 1,3,8,21,56,147,390,1027,2718,7169,18952,50025,132180,349015,921986,
%T A384641 2434831,6431386,16985525,44863652,118490229,312960192,826576635,
%U A384641 2183160062,5766102587,15229405878,40223605481,106238212160,280594628513,741103272076,1957390991519
%N A384641 Expansion of (1+2*x-x^3) / (1-x-5*x^2+x^3+2*x^4).
%C A384641 Number of walks of length n starting at vertex 2 in the following graph:
%C A384641      2
%C A384641     /|\
%C A384641  0-1 | 3
%C A384641     \|/
%C A384641      4.
%H A384641 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384641 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-1,-2).
%e A384641 a(2)=8 because we have the walks 2-1-0, 2-1-2, 2-1-4, 2-3-2, 2-3-4, 2-4-1, 2-4-2, 2-4-3.
%p A384641 a:= n-> (<<0|1|0|0|0>, <1|0|1|0|1>, <0|1|0|1|1>, <0|0|1|0|1>, <0|1|1|1|0>>^n. <<1,1,1,1,1>>)[3,1]:
%p A384641 seq(a(n), n=0..32);
%t A384641 CoefficientList[Series[(1+2*x-x^3) / (1-x-5*x^2+x^3+2*x^4), {x, 0, 32}], x]
%Y A384641 Cf. A384640 (vertices 0, 1), A384642 (vertex 3), A005824 (missing edge {2,4}), A026597 (missing edge {0,1}).
%K A384641 nonn,easy,walk
%O A384641 0,2
%A A384641 _Sean A. Irvine_, Jun 05 2025