A367621 The lexicographically earliest infinite sequence of positive numbers in which each term is a comma-child of the previous term in base 3.

1, 5, 12, 13, 18, 20, 27, 28, 32, 39, 40, 44, 51, 52, 57, 59, 67, 72, 74, 81, 82, 86, 93, 94, 98, 105, 106, 110, 117, 118, 122, 129, 130, 134, 141, 142, 146, 153, 154, 158, 166, 171, 173, 181, 186, 188, 196, 201, 203, 211, 216, 218, 226, 231, 233, 241, 245, 252

Offset: 1

Author

Michael S. Branicky and N. J. A. Sloane, Dec 20 2023

Keywords

• nonn
• base

Comments

Analogous to A367620, but with comma-children computed in base 3 (terms are shown in base 10, however).

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.

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.

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.

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000
• Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Youtube
• Michael S. Branicky, Bifurcation points on the road to infinity in the base-3 comma-child graph, starting at 1
• Giovanni Resta, Graphical representation of a portion of the graph

Crossrefs

Cf. A367355, A367618, A367619, A367620.