Partial Sums of the Fibonacci Sequence

Abstract

Let $(F_n{)}_{n\ge 1}$ be the Fibonacci sequence. Define $P(F_n):=({\sum }_{i=1}^n{F}_i{)}_{n\ge 1}$; that is, the function $P$ gives the sequence of partial sums of $(F_n)$. In this paper, we first give an identity involving $P^k(F_n)$, which is the resulting sequence from applying $P$ to $(F_n)$ $k$ times. Second, we provide a combinatorial interpretation of the numbers in $P^k(F_n)$.

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