The Pell equation \(x^{2}-(k^{2}-2)y^{2}=1\) and the corresponding integer sequence

Abstract. In [11], the second author considered the positive integer solutions of the Pell equation \(x^{2}-Dy^{2}=1\) for some specific values of \(D\) including \(D=k^{2}-2\) for an integer \(k\geq 2\). In this paper, we are able to give the \(n^{\text{th}}\) integer solution \((x_{n},y_{n})\) of \(x^{2}-(k^{2}-2)y^{2}=1\) by a different method and then we set an integer sequence \(W_{n}=pW_{n-1}-qW_{n-2}\) with parameters \(p=k^{2}-2\) and \(q=1\) and derive some algebraic relations on it.