Concatenations with binary recurrent sequences

Banks, William D. ; Luca, Florian

Journal of integer sequences, Tome 8 (2005) no. 1.

Résumé

Summary: Given positive integers $A_1,\dots ,A_t$ and $b\ge 2$, we write ${\overline{A_1\cdots A_t}}_{(b)}$ for the integer whose base-$b$ representation is the concatenation of the base-$b$ representations of $A_1,\dots ,A_t$. In this paper, we prove that if $(u_n{)}_{n\ge 0}$ is a binary recurrent sequence of integers satisfying some mild hypotheses, then for every fixed integer $t\ge 1$, there are at most finitely many nonnegative integers $n_1,\dots ,n_t$ such that ${\overline{|u_{n_1}|\cdots |u_{n_t}|}}_{(b)}$ is a member of the sequence $(|u_n|{)}_{n\ge 0}$. In particular, we compute all such instances in the special case that $b=10,t=2$, and $u_n=F_n$ is the sequence of Fibonacci numbers.

Détail

Classification : 11B37, 11B39, 11J86
Mots-clés : binary recurrent sequences, Fibonacci numbers, digits