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Journal of Integer Sequences, Vol. 28 (2025), Article 25.5.5 |
Natalya Ter-Saakov
Department of Mathematics
Rutgers University
110 Frelinghuysen Rd.
Piscataway, NJ 08854
USA
Abstract:
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.
(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.