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 A382683 #24 Jun 04 2025 17:08:50
%S A382683 1,1,3,5,13,25,59,121,273,577,1275,2733,5981,12905,28115,60849,132289,
%T A382683 286721,622739,1350613,2932109,6361209,13806923,29958441,65018001,
%U A382683 141086401,306181963,664423165,1441882653,3128970185,6790194979,14735222881,31976837633
%N A382683 Expansion of (1-x^2) / (1-x-3*x^2+x^3).
%C A382683 The number of walks of length n in the 4-vertex graph {{0,1}, {1,2}, {1,3}, {2,3}} starting at vertex 0 (see Example).
%C A382683 Also, a(n+1) is the number of such walks in the same graph starting at vertex 1.
%H A382683 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-1).
%F A382683 a(n) = A052973(n) + A052973(n-1). a(n) = A087640(n+1) - A087640(n). - _R. J. Mathar_, Jun 03 2025
%e A382683 Consider walks starting at 0 in the following graph:
%e A382683 2
%e A382683 /|
%e A382683 0-1 |
%e A382683 \|
%e A382683 3
%e A382683 The 5 walks of length 3 are 0-1-0-1, 0-1-2-1, 0-1-2-3, 0-1-3-1, and 0-1-3-2.
%p A382683 a:= n-> (<<0|1|0>, <0|0|1>, <-1|3|1>>^n. <<1,1,3>>)[1,1]:
%p A382683 seq(a(n), n=0..32); # _Alois P. Heinz_, Jun 04 2025
%t A382683 LinearRecurrence[{1,3,-1},{1,1,3},33] (* or *) CoefficientList[Series[ (1-x^2) / (1-x-3*x^2+x^3),{x,0,32}],x] (* _James C. McMahon_, Jun 02 2025 *)
%Y A382683 Cf. A087640 (walks starting at 2).
%Y A382683 Cf. A000079 (missing edge {0,1}), A108411 (missing edge {2,3}), A026581 (adding edge {0,2}), A000244 (K4).
%K A382683 nonn,walk,easy
%O A382683 0,3
%A A382683 _Sean A. Irvine_, Jun 02 2025