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 A384711 #8 Jun 08 2025 05:00:23
%S A384711 1,3,12,42,156,564,2064,7512,27408,99888,364224,1327776,4840896,
%T A384711 17648448,64342272,234575232,855204096,3117859584,11366943744,
%U A384711 41441044992,151083752448,550813774848,2008130064384,7321142777856,26691065942016,97308988551168
%N A384711 Expansion of (1+x) / (1-2*x-6*x^2).
%C A384711 Number of walks of length n in the graph K_{1,1,1,2} starting at vertex 0 when the edges are given by {{0,1}, {0,3}, {0,4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}}.
%C A384711 Also, by symmetry, the number of walks of length n starting at vertex 2 in the same graph.
%H A384711 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%F A384711 a(n) = 3*A133592(n)/2 for n>0.
%e A384711 a(2)=12 because we have the walks 0-1-0, 0-1-2, 0-1-3, 0-1-4, 0-3-0, 0-3-1, 0-3-2, 0-3-4, 0-4-0, 0-4-1, 0-4-2, 0-4-3.
%p A384711 a:= n-> (<<0|1|0|1|1>, <1|0|1|1|1>, <0|1|0|1|1>, <1|1|1|0|1>, <1|1|1|1|0>>^n. <<1, 1, 1, 1, 1>>)[1, 1]:
%p A384711 seq(a(n), n=0..32);
%t A384711 CoefficientList[Series[(1+x) / (1-2*x-6*x^2), {x, 0, 32}], x]
%Y A384711 Cf. A133592, A384712 (vertices 1, 3, 4).
%K A384711 nonn,easy,walk
%O A384711 0,2
%A A384711 _Sean A. Irvine_, Jun 07 2025