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 A384672 #11 Aug 30 2025 13:10:55
%S A384672 1,4,12,42,136,458,1512,5042,16728,55642,184840,614434,2041784,
%T A384672 6786058,22552168,74951058,249090840,827832634,2751217352,9143416194,
%U A384672 30387253880,100989154026,335627745064,1115426752498,3707013922264,12319906116890,40944028340104
%N A384672 Expansion of (1+2*x-x^2) / (1-2*x-5*x^2+2*x^3).
%C A384672 Number of walks of length n starting at vertex 1 in the following graph:
%C A384672 1---2
%C A384672 /|\ /|
%C A384672 0 | X |
%C A384672 \|/ \|
%C A384672 4---3.
%C A384672 Also, by symmetry, the number of walks of length n starting at 4 in the same graph.
%H A384672 Harvey P. Dale, <a href="/A384672/b384672.txt">Table of n, a(n) for n = 0..1000</a>
%H A384672 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384672 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,5,-2).
%e A384672 a(2)=12 because we have the walks 1-0-1, 1-0-4, 1-2-1, 1-2-3, 1-2-4, 1-3-1, 1-3-2, 1-3-4, 1-4-0, 1-4-1, 1-4-2, 1-4-3.
%p A384672 a:= n-> (<<0|1|0|0|1>, <1|0|1|1|1>, <0|1|0|1|1>, <0|1|1|0|1>, <1|1|1|1|0>>^n. <<1,1,1,1,1>>)[2,1]:
%p A384672 seq(a(n), n=0..32);
%t A384672 CoefficientList[Series[(1+2*x-x^2) / (1-2*x-5*x^2+2*x^3), {x, 0, 32}], x]
%t A384672 LinearRecurrence[{2,5,-2},{1,4,12},30] (* _Harvey P. Dale_, Aug 30 2025 *)
%Y A384672 Cf. A384671 (vertex 0), A384673 (vertices 2 and 3), A384646 (missing edge {2,4}).
%K A384672 nonn,easy,walk,changed
%O A384672 0,2
%A A384672 _Sean A. Irvine_, Jun 05 2025