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Let Vₙ(P,Q) denote the generalized Lucas sequence with parameters P and Q. For all odd relatively prime values of P and Q such that P² + 4Q > 0, we determine all indices n such that Vₙ(P,Q) = 7kx² when k|P. As an application, we determine all indices n such that the equation Vₙ = 21x² has solutions.

In this study, we determine when the Diophantine equation $x^2-kxy+y^2-2^n=0$ has an infinite number of positive integer solutions $x$ and $y$ for $0\le n\le 10.$ Moreover, we give all positive integer solutions of the same equation for $0\le n\le 10$ in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation $x^2-kxy+y^2-2^n=0$.