\(k\)-generalized Pell numbers which are concatenation of two repdigits (Q2167816)

Let \(\ge 2\) and let \((P_n^{(k)})_{n\ge -(k-2)}\) be the \(k\)-generalized Pell sequence defined by the recursion \(P_n^{(k)}=P_{n-1}^{(k)}+\cdots+P_{n-k}^{(k)}\) for \(n\ge 2\) with initial conditions \(0,0,\ldots,0,1\) (\(k-1\) zeros). They find all the members of this family of sequences which when written in base \(10\) are a concatenation of two repdigits (so, they look like \(aa\cdots abb\cdots b\), where \(a,b\in \{0,1,2,\ldots,9\}\), with \(a\ne 0\)). The largest example is \(233=P_7^{(k)}\) for all \(k\ge 6\). The proof follows the method from the paper listed as [\textit{A. Alahmadi} et al., Glas. Mat., III. Ser. 56, No. 1, 29--46 (2021; Zbl 07414938)], where the similar problem was solved with the sequence of \(k\)-generalized Fibonacci numbers instead of \(k\)-generalized Pell numbers.