The Square Terms in Generalized Lucas Sequence with Parameters $P$ And $Q$

Abstract

Let $P$ and $Q$ be nonzero integers. Generalized Lucas sequence is defined as follows: $V_{0}=2$, $V_{1}=P$, and $V_{n+1}=PV_{n}+QV_{n-1}$ for $n\geq 1$. We assume that $P$ and $Q$ are odd relatively prime integers. Firstly, we determine all indices $n$ such that $V_{n}=kx^{2}$ and $V_{n}=2kx^{2}$ when $k|P$. Then, as an application of our these results, we find all solutions of the equations $V_{n}=3x^{2}$ and $V_{n}=6x^{2}$. Moreover, we find integer solutions of some Diophantine equations.