On square numbers in the generalized Lucas sequence and generalized associated Lucas sequence (Q2888279)

The authors consider the known sequences -- the Lucas sequence and the companion Lucas sequence with parameters \(P\) and \(Q\) and generalize them to the Lucas sequence \(\{U^{(\alpha)}_n\}\) and generalized associated Lucas sequence \(\{V^{(\alpha)}_n\}\) for a fixed integer \(\alpha > 0\). They define terms of sequences \(\{U^{(\alpha)}_n\}\) and \(\{V^{(\alpha)}_n\}\) respectively by NEWLINE\[NEWLINEU^{(\alpha)}_0=0,\;U^{(\alpha)}_1=1,\;U^{(\alpha)}_n=\frac{\alpha(\alpha+1)}{2}U^{(\alpha)}_{n-1}-(\alpha+1)U^{(\alpha)}_{n-2}\;\text{ for } n \geq 2NEWLINE\]NEWLINE and NEWLINE\[NEWLINE V^{(\alpha)}_0=2,\;V^{(\alpha)}_1=\frac{\alpha(\alpha+1)}{2},\;V^{(\alpha)}_n=\frac{\alpha(\alpha+1)}{2}V^{(\alpha)}_{n-1}-(\alpha+1)V^{(\alpha)}_{n-2}\;\text{ for } n \geq 2.NEWLINE\]NEWLINE They derive results for possible square numbers in these sequences for the case \(\alpha = 3\). They also consider the generalized sequences of the type \(\{U^{(\alpha)}_n \pm 1\}\) and \(\{V^{(\alpha)}_n \pm 1\}\) and prove some elementary results regarding possible squares when \(n = 0\) and \(1\).NEWLINENEWLINEWhile reading the paper, apart from the grammatical errors, one feels that the paper could have been written in a better way.