The Comma Sequence is Finite in Other Bases

TL;DR

This paper proves that all comma sequences in bases 3 through 633 are finite and confirms a conjecture about their growth rate, showing it is approximately exponential in b log b.

Contribution

The paper provides a computational proof of finiteness for comma sequences in bases 3 to 633 and refines the growth rate estimate from exponential in b to exponential in b log b.

Findings

All comma sequences in bases 3 to 633 are finite.

Confirmed the conjecture on the growth rate of comma sequences.

Estimated the final element size as exp(O(b log b)).

Abstract

The comma sequence (1, 12, 35, 94, ...) is the lexicographically earliest sequence such that the difference of consecutive terms equals the concatenation of the digits on either side of the comma separating them. The behavior of a "generalized comma sequence" depends on the base the numbers are written in, as well as the sequence's initial values. We provide a computational proof that all comma sequences in bases 3 through 633 are finite. Relying on a combinatorial conjecture, Angelini et al. estimated that the final element of a comma sequence in base b should be roughly exp(O(b)). We prove their conjecture, but provide evidence that the correct estimate is actually exp(O(b log b)).