A summation formula for generalized $k$-bonacci numbers

TL;DR

This paper derives a summation formula for generalized k-bonacci numbers by analyzing their generating function, extending known Fibonacci number formulas to a broader class of recursive sequences.

Contribution

The paper introduces a new summation formula for k-bonacci numbers based on their generating function, generalizing Fibonacci number identities.

Findings

Derived a generating function for k-bonacci numbers

Obtained a summation formula generalizing Fibonacci identities

Extended known Fibonacci formulas to k-bonacci sequences

Abstract

In this note, we present a simple summation formula for $k$-bonacci numbers. The derivation consists in obtaining the generating function of such numbers, and noting that its evaluation at a particular value yields a formula generalizing a known expression for Fibonacci numbers.