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

TL;DR

This paper proves that, under certain conditions, only finitely many terms of a specific linear recurrence sequence are palindromic concatenations of two repdigits in a given algebraic base.

Contribution

It establishes finiteness results for sequence terms that are palindromic concatenations of repdigits in a non-unit algebraic integer base.

Findings

Finiteness of sequence terms with specified palindromic structure

Conditions under which the finiteness result holds

Application to linear recurrence sequences in algebraic bases

Abstract

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