Binary sequences meet the Fibonacci sequence

Abstract

We introduce a new family of number sequences ${\left(f\right(n\left)\right)}_{n\in N}$, governed by the recurrence relation$f\left(n\right)=af(n-u_n-1)+bf(n-u_n-2),$ where $u={\left(u_n\right)}_{n\in N}$ is a sequence with values $0,1$. Our study focuses on the properties of the sequence of quotients $h\left(n\right)=f(n+1)/f\left(n\right)$ and its set of values $V\left(f\right)=\left\{h\right(n):n\in N\}$ for various u. We give a sufficient condition for finiteness of $V\left(f\right)$ and automaticity of ${\left(h\right(n\left)\right)}_{n\in N}$, which holds in particular when u is the famous Prouhet-Thue-Morse sequence. In the automatic case, a constructive approach is used, with the help of the software Walnut. On the other hand, we prove that the set $V\left(f\right)$ is infinite for other special binary sequences u, and obtain a trichotomy in its topological type when u is eventually periodic.