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

32 pages, 10 figures