A Fibonacci type sequence with Prouhet–Thue–Morse coefficients

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 study 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 $(h_n(\mathrm{mod}m){)}_{n\in \mathbb{N}}$ for each m, and other results.

Keywords:

• Automatics sequences
• generalized Fibbonacci sequences
• linear recurrences

2020 Mathematics Subject Classifications:

• 11B39
• 11B85
• 11B37
• 11B50

Acknowledgments

We are grateful to two anonymous referees for helpful suggestions regarding the content and presentation of this work. We are also grateful to Agnieszka Dutka for proofreading and LATE X support and to Piotr Miska for useful remarks.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research of the authors was supported by the grant of the National Science Centre (NCN), Poland, no. UMO-2019/34/E/ST1/00094.