k-Generalized Pell Numbers Which are Concatenation of Two Repdigits

Abstract

Let k >= 2 and let (P-n((k)))(n >= 2-k) be the k-generalized Pell sequence defined by P-n((k)) = 2P(n-1)((k)) + P-n-2((k)) + ... + P-n-k((k)) for n >= 2 with initial conditions P--(k-2)((k)) = P--(k-3)((k)) = ... = P--1((k)) = P-0((k) )= 0, P-1((k)) = 1. In this paper, we show that 12,13, 29, 33, 34, 70,84, 88, 89, 228, and 233 are the only k-generalized Pell numbers, which are concatenation of two repdigits with at least two digits.