Meta-automatic Sequences

TL;DR

This paper introduces the concept of meta-automatic sequences, combining meta-Fibonacci and automatic recurrences, and demonstrates their properties through explicit automaton evaluations and complexity analysis.

Contribution

It defines meta-automatic sequences, constructs non-denestable examples, and analyzes their automaton representations and factor complexities.

Findings

Constructed explicit DFAO evaluations for the sequences.

Developed 4-uniform morphisms for the sequences.

Analyzed the factor complexities of the introduced sequences.

Abstract

Nested (or meta-Fibonacci) recurrences, such as the recurrence used to define Hofstadter's $Q$-sequence, along with the digit-based recurrences that underlie automatic sequences are of interest from both number-theoretic and combinatorial points of view. In this direction, Allouche and Shallit showed how the frequency sequence of a variant of the $Q$-sequence is $2$-automatic. This inspires us to introduce what may be seen as a natural combination of the recurrences for meta-Fibonacci and automatic sequences, by introducing the concept of a meta-automatic sequence. We show how it is possible to construct a meta-automatic sequence that is not denestable in terms of not being reducible, in a specific sense formalized in this paper, to certain digit-based recurrences for automatic sequences. This motivates our study of the non-denestable, meta-automatic sequences $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ introduced in this paper. For each of these integer sequences, we prove explicit DFAO evaluations, together with $4$-uniform morphisms, and we also consider the factor complexities of these sequences.