Triangular numbers in the generalized associated Pell sequence (Q1886889)

The Pell sequence \(\{P_n\}\) and the associated Pell sequence \(\{Q_n\}\) are defined by \(P_{n+2}= 2P_{n+1}+ P_n\), \(P_0= 0\), \(P_1= 1\) and \(Q_{n+2}= 2Q_{n+1}+ Q_n\), \(Q_0= Q_1= 1\) respectively, where \(n\) ranges over all integers. It is well known that \((Q_n, P_n)\) give all integer solutions of the Diophantine equations \(x^2- 2y^2= (-1)^n\). Suppose \(\alpha\) is a positive integer, then the authors generalize the two sequences to \(P^{(\alpha)}_0= 0\), \(P^{(\alpha)}_1= 1\), \(P^{(\alpha)}_{n+2}= (\alpha+1) P^{(\alpha)}_{n+1}+ {\alpha(\alpha+ 1)\over 2} P^{(\alpha)}_n\) and \(Q^{(\alpha)}_0= Q^{(\alpha)}_1= 1\), \(Q^{(\alpha)}_{n+2}= (\alpha+ 1)Q^{(\alpha)}_{n+1}+ {\alpha(\alpha+ 1)\over 2} Q^{(\alpha)}_n\) for \(n\geq 0\), and prove that each of \(Q^{(\alpha)}_0\), \(Q^{(\alpha)}_1\) and \(Q^{(\alpha)}_2\) is a triangular number (of the form \(m(m+ 1)/2\)). In the meantime, they show that when \(n\geq 3\) some \(Q^{(\alpha)}_n\) are also triangular numbers. Note that the main result (Theorem 3.13) of the paper that \(Q^{(1)}_n= Q_n\) is a triangular number if and only if \(n= 0\), \(\pm 2\) has been obtained by \textit{V. Siva Rama Prasad} and \textit{B. Srinivasa Rao} [Indian J. Pure Appl. Math. 33, No. 11, 1643--1648 (2002; Zbl 1038.11019)] with the same method. As for triangular numbers in the Pell sequence \(\{P_n\}\), the problem has been solved by \textit{W. L. McDaniel} [Fibonacci Q. 34, No. 2, 105--107 (1996; Zbl 0860.11007)].