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A384634 Expansion of (1+2*x+x^2) / (1-6*x^2-4*x^3+2*x^4).

%I A384634 #7 Jul 08 2025 11:49:43
%S A384634 1,2,7,16,48,120,338,880,2412,6392,17316,46240,124640,333920,898168,
%T A384634 2409600,6475408,17382432,46694512,125377024,336745984,904275328,
%U A384634 2428594976,6521881856,17515179200,47037120384,126321412672,339239675392,911046599168,2446649462272
%N A384634 Expansion of (1+2*x+x^2) / (1-6*x^2-4*x^3+2*x^4).
%C A384634 Number of walks of length n starting at vertex 2 in the following graph:
%C A384634      2
%C A384634     / \
%C A384634  0-1---3
%C A384634     \ /
%C A384634      4.
%H A384634 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384634 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,6,4,-2).
%e A384634 a(2)=7 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.
%p A384634 a:= n-> (<<0|1|0|0|0>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <0|1|0|1|0>>^n. <<1,1,1,1,1>>)[3,1]:
%p A384634 seq(a(n), n=0..32);
%t A384634 CoefficientList[Series[(1+2*x+x^2) / (1-6*x^2-4*x^3+2*x^4), {x, 0, 32}], x]
%Y A384634 Cf. A384633 (vertices 0, 1), A384635 (vertex 3), A005824 (missing edge {1,3}), A105476 (missing edge {1,2}).
%K A384634 nonn,easy,walk
%O A384634 0,2
%A A384634 _Sean A. Irvine_, Jun 05 2025