A367620 The lexicographically earliest infinite sequence of positive numbers in which each term is a comma-child of the previous term.

20, 22, 46, 107, 178, 260, 262, 284, 327, 401, 415, 469, 564, 610, 616, 682, 709, 807, 885, 944, 993, 1024, 1065, 1116, 1177, 1248, 1329, 1420, 1421, 1432, 1453, 1484, 1525, 1576, 1637, 1708, 1789, 1880, 1881, 1892, 1913, 1944, 1985, 2037, 2109, 2201, 2213, 2245, 2297, 2369, 2461, 2473, 2505, 2557, 2629, 2721, 2733, 2765

Offset: 1

Author

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

Keywords

• nonn
• base

Comments

Discovered by David W. Wilson in 2007 (see 2016 Angelini link).

The first choice point occurs for the term after a(412987860) = 19999999918, which has two comma-children.

We do not know which choice to take at that point. We do know by König's Infinity Lemma that 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.

Update, Dec 22 2023: We now know that the start of this sequence is one of four candidates (all other possible starts having terminated). The shortest of the four possible starts has length

8278670191169895553395510925614764265575448369172463113087634743486440833078554

In other words, we know that there are only four possibilities for the initial prefix of that length.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..20000
• Eric Angelini, The Commas Sequence, Message to Sequence Fans, Sep 06 2016. [Cached copy, with permission]
• 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, Fibonacci Quarterly 62:3 (2024), 215-232.
• Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, Local copy.
• N. J. A. Sloane, Eric Angelini's Comma Sequence, Experimental Math Seminar, Rutgers Univ., January 18, 2024, Youtube video; Slides

Crossrefs

Cf. A121805, A367338, A367346.