Partial sums of the Gibonacci sequence

TL;DR

This paper generalizes recent results on the partial sums of Fibonacci-related sequences, introduces colored Schreier sets, and offers a new combinatorial interpretation via lattice paths.

Contribution

It extends previous work by Chu on Fibonacci partial sums, introduces colored Schreier sets, and provides an alternative combinatorial interpretation.

Findings

Generalization of partial sums properties

Introduction of colored Schreier sets

New lattice path combinatorial interpretation

Abstract

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.