A Fibonacci type sequence with Prouhet-Thue-Morse coefficients (2109.15243v1)

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.