This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A384677 #11 Jul 08 2025 11:46:15
%S A384677 1,1,4,10,34,100,316,964,2992,9208,28456,87760,270928,835984,2580160,
%T A384677 7962400,24573472,75836224,234041536,722281024,2229055744,6879152512,
%U A384677 21229965952,65518430464,202198419712,624010629376,1925778076672,5943201831424,18341494710784
%N A384677 Expansion of (1-x-2*x^2) / (1-2*x-4*x^2+2*x^3).
%C A384677 Number of walks of length n starting at vertex 0 in the following graph:
%C A384677 2
%C A384677 /|\
%C A384677 0-1-+-3
%C A384677 \|/
%C A384677 4.
%C A384677 Also, for n>=1, the number of walks of length n-1 starting at vertex 1 in the same graph.
%H A384677 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384677 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,4,-2).
%e A384677 a(3)=10 because we have the walks 0-1-0-1, 0-1-2-1, 0-1-2-3, 0-1-2-4, 0-1-3-1, 0-1-3-2, 0-1-3-4, 0-1-4-1, 0-1-4-2, 0-1-4-3.
%p A384677 a:= n-> (<<0|1|0|0|0>, <1|0|1|1|1>, <0|1|0|1|1>, <0|1|1|0|1>, <0|1|1|1|0>>^n. <<1,1,1,1,1>>)[1,1]:
%p A384677 seq(a(n), n=0..32);
%t A384677 CoefficientList[Series[(1-x-2*x^2) / (1-2*x-4*x^2+2*x^3), {x, 0, 32}], x]
%t A384677 Table[(MatrixPower[{{0,1,0,0,0},{1,0,1,1,1},{0,1,0,1,1},{0,1,1,0,1},{0,1,1,1,0}},n].{1,1,1,1,1}),{n,0,28}][[All,1]] (* _Shenghui Yang_, Jun 07 2025 *)
%Y A384677 Cf. A384678 (vertices 2, 3, 4), A000244 (missing edge {0,1}), A384633 (missing edge {2,4}), A384640 (missing edge {1,3}).
%K A384677 nonn,easy,walk
%O A384677 0,3
%A A384677 _Sean A. Irvine_, Jun 05 2025