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 A384598 #16 Jun 04 2025 18:03:01
%S A384598 1,1,2,4,8,18,38,86,186,418,914,2042,4490,9994,22042,48954,108154,
%T A384598 239898,530522,1175898,2601882,5764634,12759322,28262298,62566554,
%U A384598 138567834,306790810,679404442,1504298906,3331199386,7376004506,16333395354,36166416794
%N A384598 Expansion of (1-3*x^2) / (1-x-4*x^2+2*x^3+2*x^4).
%C A384598 Number of walks of length n on the following graph starting at vertex 0:
%C A384598 3
%C A384598 /|
%C A384598 0-1-2 |
%C A384598 \|
%C A384598 4.
%C A384598 Also, for n>=1, the number of walks of length n-1 starting from vertex 1 in the same graph.
%H A384598 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%H A384598 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-2,-2).
%e A384598 a(3)=4 because we have the walks 0-1-0-1, 0-1-2-1, 0-1-2-3, 0-1-2-4.
%p A384598 a:= n-> (<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>, <-2|-2|4|1>>^n. <<1,1,2,4>>)[1,1]:
%p A384598 seq(a(n), n=0..32); # _Alois P. Heinz_, Jun 04 2025
%t A384598 CoefficientList[Series[(1 - 3*x^2)/(1 - x - 4*x^2 + 2*x^3 + 2*x^4), {x, 0, 32}], x] (* _Michael De Vlieger_, Jun 04 2025 *)
%Y A384598 Cf. A384599 (vertex 2), A384600 (vertex 3), A062112 (missing edge {3,4}), A382683 (missing edge {0,1}).
%K A384598 nonn,walk,easy
%O A384598 0,3
%A A384598 _Sean A. Irvine_, Jun 04 2025