Repdigits in generalized Pell sequences

Abstract

For an integer $k\ge 2$, let $({n})_n$ be the $k-$generalized Pell sequence which starts with $0,\ldots ,0,1$ ($k$ terms) and each term afterwards is given by the linear recurrence ${n} = 2{n-1}+{n-2}+\cdots +{n-k}$. In this paper, we find all $k$-generalized Pell numbers with only one distinct digit (the so-called repdigits). Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for repdigits in the usual Pell sequence $(P_n^{(2)})_n$.