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 A384646 #10 Jul 08 2025 11:56:10
%S A384646 1,2,7,19,58,167,495,1446,4255,12475,36642,107527,315687,926606,
%T A384646 2720095,7984499,23438186,68800871,201960799,592841526,1740247263,
%U A384646 5108376491,14995295858,44017672839,129210905111,379289861022,1113379732255,3268250847587,9593729230906
%N A384646 Expansion of (1+x) / (1-x-5*x^2-2*x^3).
%C A384646 Number of walks of length n starting at vertex 0 in the following graph:
%C A384646 1---2
%C A384646 /|\ |
%C A384646 0 | \ |
%C A384646 \| \|
%C A384646 4---3.
%C A384646 Also, by symmetry, the number of walks of length n starting at vertex 2 in the same graph.
%H A384646 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384646 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,2).
%F A384646 a(n) = A353964(n)+A353964(n-1). - _R. J. Mathar_, Jun 07 2025
%e A384646 a(2)=7 because we have the walks 0-1-0, 0-1-2, 0-1-3, 0-1-4, 0-4-0, 0-4-1, 0-4-3.
%p A384646 a:= n-> (<<0|1|0|0|1>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <1|1|0|1|0>>^n. <<1,1,1,1,1>>)[1,1]:
%p A384646 seq(a(n), n=0..32);
%t A384646 CoefficientList[Series[(1+x) / (1-x-5*x^2-2*x^3), {x, 0, 32}], x]
%Y A384646 Cf. A384647 (vertex 1), A384648 (vertices 3 and 4), A077937 (missing edge {1,3}).
%K A384646 nonn,easy,walk
%O A384646 0,2
%A A384646 _Sean A. Irvine_, Jun 05 2025