Journal of Integer Sequences, Vol. 28 (2025), Article 25.5.5

The Comma Sequence is Finite in Other Bases


Robert Dougherty-Bliss
Department of Mathematics
Dartmouth College
27 N. Main St.
Hanover, NH 03755
USA

Natalya Ter-Saakov
Department of Mathematics
Rutgers University
110 Frelinghuysen Rd.
Piscataway, NJ 08854
USA

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 in which the numbers are written, as well as the sequence's initial values. We give a computational proof that all comma sequences in bases 3 through 633 are finite.

Angelini et al. conjectured a generating function formula related to the comma sequence and, from this, predicted that the final element of a comma sequence in base b should be roughly exp(O(b)). We give a combinatorial proof of Angelini's conjecture, but also numerical evidence that the prediction about the final element is wrong. We provide a new random model for the comma sequence that predicts a final term of exp(O(b log b)), which aligns with our simulations.


Full version:  pdf,    ps,    latex    


(Concerned with sequences A121805 A136107 A330128.)


Received January 6 2025; revised versions received January 7 2025; August 23 2025. Published in Journal of Integer Sequences, September 19 2025.


Return to Journal of Integer Sequences home page