# Binary patterns in the Prouhet-Thue-Morse sequence

## TL;DR

This paper characterizes binary patterns in the Prouhet-Thue-Morse sequence, showing most are segments of the sequence except for two specific patterns, and identifies patterns arising from non-trivial endomorphisms.

## Contribution

It provides a comprehensive classification of binary patterns in the sequence, including those generated by non-trivial endomorphisms, clarifying previous attributions.

## Key findings

- All binary patterns except two are segments of the sequence.
- Identified finite patterns generated by non-trivial endomorphisms.
- Clarified historical attribution of the pattern classification.

## Abstract

We show that, with the exception of the words $a^2ba^2$ and $b^2ab^2$, all (finite or infinite) binary patterns in the Prouhet-Thue-Morse sequence can actually be found in that sequence as segments (up to exchange of letters in the infinite case). This result was previously attributed to unpublished work by D. Guaiana and may also be derived from publications of A. Shur only available in Russian. We also identify the (finitely many) finite binary patterns that appear non trivially, in the sense that they are obtained by applying an endomorphism that does not map the set of all segments of the sequence into itself.