A fast recurrence for Fibonacci and Lucas numbers

TL;DR

This paper introduces a simplified and efficient double recurrence relation for Fibonacci and Lucas numbers, enabling faster computation, with the recurrence likely being novel despite its simplicity.

Contribution

It presents a new, simple double recurrence relation for Fibonacci and Lucas numbers that improves computational speed.

Findings

The recurrence enables faster Fibonacci number calculations.

The recurrence is simple to implement and efficient.

It appears to be a novel relation not previously documented.

Abstract

We derive the double recurrence $e_n = \frac{1}{2}(a_{n-1}+5b_{n-1}); f_{n} = \frac{1}{2}(a_{n-1}+b_{n-1})$ with $e_0=2;f_0=0$ for the Fibonacci numbers, leading to an extremely simple and fast implementation. Though the recurrence is probably not new, we have not been able to find a reference for it.