A Fibonacci type sequence with Prouhet-Thue-Morse coefficients

TL;DR

This paper introduces a new Fibonacci-like sequence based on Prouhet-Thue-Morse coefficients, exploring its arithmetic properties, automaticity, and non-vanishing behavior, thus extending the understanding of sequences related to binary digit sums.

Contribution

It defines a novel recursive sequence involving Prouhet-Thue-Morse coefficients and investigates its arithmetic properties, automaticity, and non-vanishing characteristics.

Findings

Sequence $(h_n)$ is non-zero for $n \,\geq\, 5$

Sequence $(h_n \,\bmod\, m)$ is automatic for each m

Several arithmetic properties of $(h_n)$ are established

Abstract

Let $t_n = (-1)^{s_2(n)}$, where $s_2(n)$ is the sum of binary digits function. The sequence $(t_n)_{n\in \mathbb N}$ is the well-known Prouhet-Thue-Morse sequence. In this note we initiate the study of the sequence $(h_n)_{n\in \mathbb N}$, where $h_0 = 0, h_1 = $1 and for $n \ge 2$ we define $h_n$ recursively as follows:$ h_n = t_n h_{n-1} + h_{n-2}$. We prove several results concerning arithmetic properties of the sequence $(h_n )_{n\in \mathbb N}$. In particular, we prove non-vanishing of $h_n$ for $n \ge 5$, automaticity of the sequence $(h_n \pmod m)_{n\in \mathbb N}$ for each m, and other results.