For Lucas sequences U_n(x,±1) the equation ∑_{j=1}^k j U_j^p = U_n^q has only the listed solutions when k=2 (any p,q) and none when k≥3 and max{p,q}≤11 under mild conditions on x.
Hajdu,On a conjecture of Sch¨ affer concerning the equation1 k +· · ·+x k =y n, J
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Extension of the Equation $\sum\limits_{j=1}^{k}jF_{j}^{p}=F_{n}^{q}$ to a Family of Lucas Sequences
For Lucas sequences U_n(x,±1) the equation ∑_{j=1}^k j U_j^p = U_n^q has only the listed solutions when k=2 (any p,q) and none when k≥3 and max{p,q}≤11 under mild conditions on x.