A fast recurrence for Fibonacci and Lucas numbers

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.