Repdigits in k-generalized Pell sequence

TL;DR

This paper investigates when generalized Pell sequence numbers are repdigits, concluding only two specific sequence terms, 33 and 88, are repdigits with at least two digits.

Contribution

It characterizes all repdigit numbers within the k-generalized Pell sequence, identifying only two such terms.

Findings

Only P_5^{(3)}=33 and P_6^{(4)}=88 are repdigits in the sequence.

Repdigits with at least two digits do not occur for other sequence terms.

The study solves a specific Diophantine equation related to repdigits in the sequence.

Abstract

Let $k\geq 2$ and let $(P_{n}^{(k)})_{n\geq 2-k}$ be $k$-generalized Pell sequence defined by \begin{equation*}P_{n}^{(k)}=2P_{n-1}^{(k)}+P_{n-2}^{(k)}+...+P_{n-k}^{(k)}\end{equation*} for $n\geq 2$ with initial conditions \begin{equation*}P_{-(k-2)}^{(k)}=P_{-(k-3)}^{(k)}=\cdot \cdot \cdot =P_{-1}^{(k)}=P_{0}^{(k)}=0,P_{1}^{(k)}=1. \end{equation*} In this paper, we deal with the Diophantine equation \begin{equation*}P_{n}^{(k)}=d\left( \frac{10^{m}-1}{9}\right)\end{equation*} in positive integers $n,m,k,d$ with $k\geq 2,$ $m\geq 2$ and $1\leq d\leq 9$. We will show that repdigits with at least two digits in the sequence $\left( P_{n}^{(k)}\right)_{n\geq 2-k}$ are the numbers\ $P_{5}^{(3)}=33$ and $P_{6}^{(4)}=88.$