Composition inverses of the variations of the Baum-Sweet sequence

TL;DR

This paper investigates the arithmetic properties of formal inverses of sequences related to the Baum-Sweet sequence, revealing recurrence relations, regularity of certain index sequences, and unexpected links to the Thue-Morse sequence.

Contribution

It introduces new recurrence relations for the formal inverses of Baum-Sweet related sequences and analyzes their regularity and connections to other well-known sequences.

Findings

Derived recurrence relations for the formal inverses.

Determined the regularity of index sequences where inverses take specific values.

Discovered an unexpected connection to the Thue-Morse sequence.

Abstract

Studying and comparing arithmetic properties of a given automatic sequence and the sequence of coefficients of the composition inverse of the associated formal power series (the formal inverse of that sequence) is an interesting problem. This problem was studied before for the Thue-Morse sequence. In this paper, we study arithmetic properties of the formal inverses of two sequences closely related to the well-known Baum-Sweet sequence. We give the recurrence relations for their formal inverses and we determine whether the sequences of indices at which these formal inverses take value $0$ and $1$ are regular. We also show an unexpected connection between one of the obtained sequences and the formal inverse of the Thue-Morse sequence.