Sums of $k$-bonacci Numbers

TL;DR

This paper provides a combinatorial proof for a formula expressing partial sums of the $k$-bonacci sequence as alternating sums involving binomial coefficients and powers of two, also deriving a new formula for the sequence itself.

Contribution

It introduces a novel combinatorial proof for the partial sums formula of the $k$-bonacci sequence and derives a new explicit formula for the sequence.

Findings

Partial sums of $k$-bonacci sequence expressed as alternating sums of binomial coefficients and powers of two.

Derived a new explicit formula for the $k$-bonacci numbers.

Provided combinatorial proof techniques for sequence identities.

Abstract

We give a combinatorial proof of a formula giving the partial sums of the $k$-bonacci sequence as alternating sums of powers of two multiplied by binomial coefficients. As a corollary we obtain a formula for the $k$-bonacci numbers.