Evaluation of weighted Fibonacci sums of a certain type

TL;DR

This paper derives a formula for evaluating weighted sums involving generalized Fibonacci numbers, providing explicit examples and expanding the understanding of Fibonacci sum evaluations.

Contribution

It introduces a new formula for weighted generalized Fibonacci sums of a specific type, with several explicit evaluations as examples.

Findings

Derived a general formula for weighted Fibonacci sums

Provided explicit evaluations for specific cases

Enhanced methods for Fibonacci sum calculations

Abstract

We derive a formula for the evaluation of weighted generalized Fibonacci sums of the type $S_k^n (w,r) = \sum_{j = 0}^k {w^j j^r G_j{}^n }$. Several explicit evaluations are presented as examples.