Linear recurrence sequences and palindromic concatenations of two repdigits in base $\beta$

Abstract

Let $\beta$ be a non-unit real algebraic integer greater than one and $\{a_n{\}}_{n\ge 0}$ be a sequence satisfying a linear recurrence relation $a_{n+3}=a{a}_{n+2}+b{a}_{n+1}+c{a}_n$. Under certain conditions, we prove that the number of $a_n$ which are palindromic concatenations of two repdigits in base $\beta$ is finite.