The Look-and-Say The Biggest Sequence Eventually Cycles

TL;DR

This paper introduces a variant of Conway's sequence called 'look-and-say the biggest' (LSB), proving that unlike the original, LSB always reaches a cycle with a maximum period of 9, regardless of the starting value.

Contribution

The paper proves that the LSB sequence always reaches a cycle and characterizes the maximum cycle length, contrasting with the exponential growth of Conway's original sequence.

Findings

LSB sequence always reaches a cycle for any starting value.

All cycles in LSB have a period of at most 9.

Contrasts with Conway's sequence which grows exponentially.

Abstract

In this paper we consider a variant of Conway's sequence (OEIS A005150, A006715) defined as follows: the next term in the sequence is obtained by considering contiguous runs of digits, and rewriting them as $ab$ where $b$ is the digit and $a$ is the maximum of $b$ and the run's length. We dub this the "look-and-say the biggest" (LSB) sequence. Conway's sequence is very similar ($b$ is just the run's length). For any starting value except 22, Conway's sequence grows exponentially: the ration of lengths converges to a known constant $\lambda$. We show that LSB does not: for every starting value, LSB eventually reaches a cycle. Furthermore, all cycles have a period of at most 9.