2020/06/11 by Éric Brier, Rémi Géraud, Brier, Éric +7
Computer Science · Mathematics · #Benford’s Law and Fraud Detection #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL) #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2006.06837
openalex publication_date 2020/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A look-and-say sequence is obtained iteratively by reading off the digits of the current value, grouping identical digits together: starting with 1, the sequence reads: 1, 11, 21, 1211, 111221, 312211, etc. (OEIS A005150). Starting with any digit d ≠ 1 gives Conway's sequence: d, 1d, 111d, 311d, 13211d, etc. (OEIS A006715). Conway popularised these sequences and studied some of their properties. In this paper we consider a variant subbed "look-and-say again" where digits are repeated twice. We prove that the look-and-say again sequence contains only the digits 1, 2, 4, 6, d, where d represents the starting digit. Such sequences decompose and the ratio of successive lengths converges to Conway's constant. In fact, these properties result from a commuting diagram between look-and-say again sequences and "classical" look-and-say sequences. Similar results apply to the "look-and-say three times" sequence.