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A384673 Expansion of (1+x) / (1-2*x-5*x^2+2*x^3).

%I A384673 #6 Aug 31 2025 08:58:30
%S A384673 1,3,11,35,119,391,1307,4331,14415,47871,159155,528835,1757703,
%T A384673 5841271,19413387,64517723,214419839,712601519,2368266787,7870701491,
%U A384673 26157533879,86932041639,288910349691,960165839819,3191019344815,10605047189343,35244859423123
%N A384673 Expansion of (1+x) / (1-2*x-5*x^2+2*x^3).
%C A384673 Number of walks of length n starting at vertex 2 in the following graph:
%C A384673     1---2
%C A384673    /|\ /|
%C A384673   0 | X |
%C A384673    \|/ \|
%C A384673     4---3.
%C A384673 Also, by symmetry, the number of walks of length n starting at 3 in the same graph.
%H A384673 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384673 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,5,-2).
%e A384673 a(2)=11 because we have the walks 2-1-0, 2-1-2, 2-1-3, 2-1-4, 2-3-1, 2-3-2, 2-3-4, 2-4-0, 2-4-1, 2-4-2, 2-4-3.
%p A384673 a:= n-> (<<0|1|0|0|1>, <1|0|1|1|1>, <0|1|0|1|1>, <0|1|1|0|1>, <1|1|1|1|0>>^n. <<1,1,1,1,1>>)[3,1]:
%p A384673 seq(a(n), n=0..32);
%t A384673 CoefficientList[Series[(1+x) / (1-2*x-5*x^2+2*x^3), {x, 0, 32}], x]
%Y A384673 Cf. A384671 (vertex 0), A384672 (vertices 1 and 4), A384646 (missing edge {2,4}).
%K A384673 nonn,easy,walk,changed
%O A384673 0,2
%A A384673 _Sean A. Irvine_, Jun 05 2025