Partial Sums of the Fibonacci Sequence

TL;DR

This paper investigates the properties of partial sums of the Fibonacci sequence, deriving identities for iterated sums and offering a combinatorial interpretation of these sums.

Contribution

It introduces new identities for iterated partial sums of Fibonacci numbers and provides a combinatorial perspective on these sequences.

Findings

Derived identities involving $P^k(F_n)$

Provided a combinatorial interpretation of the partial sums

Enhanced understanding of Fibonacci partial sums

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 $P^k(F_n)$.