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A367621
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The lexicographically earliest infinite sequence of positive numbers in which each term is a comma-child of the previous term in base 3.
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1
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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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
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CROSSREFS
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KEYWORD
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nonn,base
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AUTHOR
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STATUS
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approved
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