Some polynomial values binary recurrences

We study the diophantine equations G_n = (^𝓍_k)and G_n = ∑^x _i=1  i^k in integers n ≥ 0 and x ≥  k (x ≥ 1) with fixed positive integer k, where G_n denotes the n^th term of a binary recurrence of certain types. We give a simple practical algorithm which can solve the first equation if k = 3 and the second equation if k = 2. As a demonstration, this algorithm is applied to provide the solutions of the second equation in case of the Fibonacci, Lucas and Pell sequences. Further, using different methods, the problems G_n = (^𝓍₄)  and G_n = ∑^x _i=1  i³  will be solved for the three famous recurrences.