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 $0,\dots ,0,0,1$, 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.

arXiv comment14 pages