Binary sequences meet the Fibonacci sequence

TL;DR

This paper introduces a new family of meta-Fibonacci sequences governed by a recurrence involving a binary sequence, analyzing their quotient sequences and the conditions for finiteness or infiniteness of their value sets, with applications to automatic sequences.

Contribution

It establishes conditions for the finiteness and automaticity of the quotient sequence set, especially for sequences derived from the Prouhet-Thue-Morse sequence, and explores the topological nature of these sets.

Findings

Finiteness of the value set is characterized by a specific condition.

Automaticity of the quotient sequence is proven for certain binary sequences.

The value set can be infinite or have a specific topological structure depending on the binary sequence.

Abstract

We introduce a new family of meta-Fibonacci sequences $(f(n))_{n\in\mathbb{N}}$, governed by the recurrence relation $$f(n)=af(n-u_{n}-1)+bf(n-u_{n}-2),$$ where $\mathbf{u}=(u_{n})_{n\in \mathbb{N}}$ is a sequence with values $0,1$. Our study focuses on the properties of the sequence of quotients $h(n) = f(n+1)/f(n)$ and its set of values $\mathcal{V}(f)=\{h(n): n \in \mathbb{N}\}$ for various $\mathbf{u}$. We give a sufficient condition for finiteness of $\mathcal{V}(f)$ and automaticity of $(h(n))_{n \in \mathbb{N}}$, which holds in particular when $\mathbf{u}$ is the famous Prouhet-Thue-Morse sequence. In the automatic case, a constructive approach is used, with the help of the software \texttt{Walnut}. On the other hand, we prove that the set $\cal{V}(f)$ is infinite for other special binary sequences $\mathbf{u}$, and obtain a trichotomy in its topological type when $\mathbf{u}$ is eventually periodic.