On $k$-Pell numbers that are Palindromes formed by two distinct Repdigits

Abstract

Let $k \ge 2$ and consider the sequence $\{P_n^{(k)}\}_{n \ge 2-k}$ of $k$-generalized Pell numbers, which begins with the first $k$ terms as $0, \ldots, 0, 0, 1$, and satisfies the recurrence relation $P_n^{(k)} = 2P_{n-1}^{(k)} + P_{n-2}^{(k)} + \cdots + P_{n-k}^{(k)}$ for all $n \ge 2$. In this work, we identify all terms in the $k$-Pell sequence that can be expressed as palindromes formed by concatenating two distinct repdigits.