A simplified Binet formula for k-generalized Fibonacci numbers

TL;DR

This paper introduces a simplified Binet-style formula for k-generalized Fibonacci numbers, enabling easy computation by rounding to the nearest integer, and discusses concurrent independent discoveries.

Contribution

It provides a new, simplified Binet formula for k-generalized Fibonacci numbers that requires only rounding to the nearest integer to generate the sequence.

Findings

The formula accurately produces k-generalized Fibonacci numbers.

The method simplifies computation of these sequences.

Independent discovery by other researchers was noted.

Abstract

We present a particularly nice Binet-style formula that can be used to produce the k-generalized Fibonacci numbers (that is, the Tribonaccis, Tetranaccis, etc). Furthermore, we show that in fact one needs only take the integer closest to the first term of this Binet-style formula to generate the desired sequence. These results were also found (independently) at about the same time by Zhaohui Du of Singapore, China. We are working on a joint paper.