Journal of Integer Sequences, Vol. 8 (2005), Article 05.1.3

Concatenations with Binary Recurrent Sequences

William D. Banks
Department of Mathematics
University of Missouri
Columbia, MO 65211

Florian Luca
Instituto de Matemáticas
Universidad Nacional Autónoma de México
C.P. 58180
Morelia, Michoacán

Abstract: Given positive integers $A_1,\ldots,A_t$ 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 $A_1,\ldots,A_t$. 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 $n_1,\ldots,n_t$ such that ${\overline{\vert u_{n_1}\vert\cdots \vert u_{n_t}\vert}}_{\,(b)}$ is a member of the sequence $(\vert u_n\vert)_{n\ge 0}$. 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.

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Received August 30 2004; revised version received January 13 2005. Published in Journal of Integer Sequences January 14 2005. Small revisions, January 17 2005.

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