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The sums $\sum_{j = 0}^k {u_{rj + s}^{2n}z^j }$, $\sum_{j = 0}^k {u_{rj +s}^{2n-1}z^j }$, $\sum_{j = 0}^k {v_{rj + s}^{n}z^j }$ and $\sum_{j = 0}^k{w_{rj + s}^{n}z^j }$ are evaluated; where $n$ is any positive integer, $r$,$s$ and $k$ are any arbitrary integers, $z$ is arbitrary, $(u_i)$ and $(v_i)$are the Lucas sequences of the first kind, and of the second kind,respectively; and $(w_i)$ is the Horadam sequence. Pantelimon St\uanic\ua setout to evaluate the sum $\sum_{j = 0}^k {w_j^n z^j }$. His solution is notcomplete because he made the assumption that $w_0=0$, thereby givingeffectively only the partial sum for $(u_i)$, the Lucas sequence of the firstkind.

Partial sums and generating functions for powers of second order sequences with indices in arithmetic progression