Sum of terms of recurrence sequences in the solution sets of generalized Pell equations

Let $(X_{k})_{k\geq 1}$ and $(Y_k)_{k\geq 1}$ be sequences of $X$ and $Y$-coordinates of the positive integer solutions $(x,y)$ of the equation $x^2 - dy^2 = t$. In this paper, we completely describe these recurrence sequences such that the sums of two terms of recurrence sequences in the solution sets of generalized Pell equations are infinitely many. Further, we give an upper bound for the number of such terms when there are only finitely many of them. This work is motivated by the recent paper of Hajdu and Sebestyén (Int. J. Number Theory 18 (2022), 1605-1612).