On the additive complexity of a Thue-Morse like sequence

TL;DR

This paper analyzes the additive complexity of a Thue-Morse like sequence generated by a specific morphism, revealing an exact logarithmic formula and its regularity properties.

Contribution

It provides a precise formula for the additive complexity of a generalized Thue-Morse sequence and establishes its 2-regularity.

Findings

Additive complexity follows a logarithmic pattern: 2*floor(log2(n))+3.

The sequence of additive complexities is 2-regular.

The formula applies for all integers n ≥ 1.

Abstract

In this paper, we study the additive complexity $\rho^{+}_{\mathbf{t}}(n)$ of a Thue-Morse like sequence $\mathbf{t}=\sigma^{\infty}(0)$ with the morphism $\sigma: 0\to 01, 1\to 12, 2\to 20$. We show that $\rho^{+}_{\mathbf{t}}(n)=2\lfloor\log_2(n)\rfloor+3$ for all integers $n\geq 1$. Consequently, $(\rho_{\mathbf{t}}(n))_{n\geq 1}$ is a $2$-regular sequence.