  Recently, Chu studied some properties of the partial sums of the sequence
$P^k(F_n)$, where $P(F_n)=\big(\sum_{i=1}^nF_i\big)_{n\geq1}$ and
$(F_n)_{n\geq1}$ is the Fibonacci sequence, and gave its combinatorial
interpretation. We generalize those results, introduce colored Schreier sets,
and give another equivalent combinatorial interpretation by means of lattice
path.
