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A384611 Expansion of (1+2*x-x^3) / (1-5*x^2+2*x^4).

%I A384611 #6 Jun 04 2025 23:20:58
%S A384611 1,2,5,9,23,41,105,187,479,853,2185,3891,9967,17749,45465,80963,
%T A384611 207391,369317,946025,1684659,4315343,7684661,19684665,35053987,
%U A384611 89792639,159900613,409593865,729395091,1868384047,3327174229,8522732505,15177080963,38876894431
%N A384611 Expansion of (1+2*x-x^3) / (1-5*x^2+2*x^4).
%C A384611 Number of walks of length n starting at vertex 2 in the following graph:
%C A384611      2
%C A384611     / \
%C A384611  0-1   3
%C A384611     \ /
%C A384611      4.
%H A384611 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384611 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,5,0,-2).
%e A384611 a(3)=9 because we have the walks 2-1-0-1, 2-1-2-1, 2-1-2-3, 2-1-4-1, 2-1-4-3, 2-3-2-1, 2-3-2-3, 2-3-4-1, 2-3-4-3.
%p A384611 a:= n-> (<<0|1|0|0|0>, <1|0|1|0|1>, <0|1|0|1|0>, <0|0|1|0|1>, <0|1|0|1|0>>^n. <<1,1,1,1,1>>)[3,1]:
%p A384611 seq(a(n), n=0..32);
%t A384611 CoefficientList[Series[(1+2*x-x^3) / (1-5*x^2+2*x^4), {x, 0, 32}], x]
%Y A384611 Cf. A005824 (vertices 0, 1), A079162 (vertex 3), A000079 (missing edge {0,1}).
%K A384611 nonn,easy,walk
%O A384611 0,2
%A A384611 _Sean A. Irvine_, Jun 04 2025