Binary sequences meet the Fibonacci sequence

TL;DR

This work investigates meta-Fibonacci sequences defined by $f(n)=a f(n-u_n-1)+b f(n-u_n-2)$ with a binary index $u_n$, focusing on the quotient $h(n)=f(n+1)/f(n)$ and the set of quotients $\mathcal{V}(f)$. It develops both analytic and automata-theoretic tools to characterize when $\mathcal{V}(f)$ is finite and when $(h(n))$ is automatic, with the PTM sequence providing a central, fully worked case in which $\mathcal{V}(f)$ has exactly seven elements and $h(n)$ is $2$-automatic. The paper further generalizes to two-sequence recurrences, proves bounded-gap conditions guarantee finiteness and automaticity, and presents a constructive Walnut-based framework for building associated automata. It also establishes broad results on the infinitude and topological structure of $\mathcal{V}(f)$ for sequences $u$ with long $010^d$ blocks and for eventually periodic $u$, including a trichotomy that mirrors classical Kepler-limit behavior. Together, these results illuminate how binary automatic patterns govern the arithmetic and dynamical properties of meta-Fibonacci quotients and provide computational methods to analyze such recurrences.

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.

Figures (3)

• Figure 1: A DFAO generating the PTM sequence
• Figure 2: A $2$-DFAO generating the sequence $(k_n)_{n \in \mathbb{N}}$
• Figure 3: Plot of $f(n+1)/f(n)$ for $n\in\{1,\ldots,2^{12}\}$

Theorems & Definitions (25)

• Theorem 3.1
• proof
• Remark 3.2
• Remark 3.3
• Theorem 4.1
• proof
• Proposition 4.2
• Remark 4.3
• Proposition 4.4
• proof