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 A367621 #28 Feb 07 2024 13:21:43
%S A367621 1,5,12,13,18,20,27,28,32,39,40,44,51,52,57,59,67,72,74,81,82,86,93,
%T A367621 94,98,105,106,110,117,118,122,129,130,134,141,142,146,153,154,158,
%U A367621 166,171,173,181,186,188,196,201,203,211,216,218,226,231,233,241,245,252
%N A367621 The lexicographically earliest infinite sequence of positive numbers in which each term is a comma-child of the previous term in base 3.
%C A367621 Analogous to A367620, but with comma-children computed in base 3 (terms are shown in base 10, however).
%C A367621 We know from A367619 that the comma-child graph in base 3, starting at 1, is an infinite tree rooted at 1. By König's Infinity Lemma, an infinite path in that graph exists and hence this sequence is well defined for all n. Therefore, at any bifurcation point, one or both forks will extend to infinity. The definition of this sequence requires that we choose the smallest fork that has an infinite continuation.
%C A367621 The terms in the data and b-file include a number of bifurcation points, but in each case the path chosen was the only one that did not lead to a finite sequence; see linked a-file.
%C A367621 We conjecture that choosing down-up-down-up-... is an infinite path, visiting the base-3 terms 1 2^{1+4*j} then 2 0^{2+4*j} for j in 0..oo, where ^ denotes repeated concatenation. This has been tested empirically up to j = 4300.
%H A367621 Michael S. Branicky, <a href="/A367621/b367621.txt">Table of n, a(n) for n = 1..10000</a>
%H A367621 Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, <a href="http://arxiv.org/abs/2401.14346">arXiv:2401.14346</a>, <a href="https://www.youtube.com/watch?v=_EHAdf6izPI">Youtube</a>
%H A367621 Michael S. Branicky, <a href="/A367621/a367621.txt">Bifurcation points on the road to infinity in the base-3 comma-child graph, starting at 1</a>
%H A367621 Giovanni Resta, <a href="/A367621/a367621.pdf">Graphical representation of a portion of the graph</a>
%Y A367621 Cf. A367355, A367618, A367619, A367620.
%K A367621 nonn,base
%O A367621 1,2
%A A367621 _Michael S. Branicky_ and _N. J. A. Sloane_, Dec 20 2023