Stuttering Conway Sequences Are Still Conway Sequences

TL;DR

This paper investigates a variant of the look-and-say sequence called "look-and-say again," proving it contains only specific digits, decomposes predictably, and converges to Conway's constant, with similar results for "look-and-say three times."

Contribution

It introduces and analyzes the "look-and-say again" sequence, establishing its digit set, decomposition properties, and convergence behavior, extending Conway sequence theory.

Findings

Sequences contain only digits 1, 2, 4, 6, and the starting digit.

The ratio of successive sequence lengths converges to Conway's constant.

Similar properties hold for the "look-and-say three times" sequence.

Abstract

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 $d \neq 1$ 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.