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A384640 Expansion of (1-3*x^2) / (1-x-5*x^2+x^3+2*x^4).

%I A384640 #7 Jul 08 2025 11:52:09
%S A384640 1,1,3,7,19,49,131,343,911,2397,6347,16735,44251,116785,308611,814815,
%T A384640 2152583,5684477,15015355,39655527,104742659,276635985,730663043,
%U A384640 1929789255,5096983167,13461994429,35555794923,93909205391,248032219243,655098462417,1730238763395
%N A384640 Expansion of (1-3*x^2) / (1-x-5*x^2+x^3+2*x^4).
%C A384640 Number of walks of length n starting at vertex 0 in the following graph:
%C A384640      2
%C A384640     /|\
%C A384640  0-1 | 3
%C A384640     \|/
%C A384640      4.
%C A384640 Also, for n>=1, the number of walks of length n-1 starting at vertex 1 in the same graph.
%H A384640 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384640 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-1,-2).
%e A384640 a(3)=7 because we have the walks 0-1-0-1, 0-1-2-1, 0-1-2-3, 0-1-2-4, 0-1-4-1, 0-1-4-2, 0-1-4-3.
%p A384640 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>>)[1,1]:
%p A384640 seq(a(n), n=0..32);
%t A384640 CoefficientList[Series[(1-3*x^2) / (1-x-5*x^2+x^3+2*x^4), {x, 0, 32}], x]
%Y A384640 Cf. A384641 (vertex 2), A384642 (vertex 3), A005824 (missing edge {2,4}), A026597 (missing edge {0,1}).
%K A384640 nonn,easy,walk
%O A384640 0,3
%A A384640 _Sean A. Irvine_, Jun 05 2025