The golden ratio, Fibonacci numbers and BBP-type formulas

Abstract

We derive interesting arctangent identities involving the golden ratio, Fibonacci numbers and Lucas numbers. Binary BBP-type formulas for the arctangents of certain odd powers of the golden ratio are also derived, for the first time in the literature. Finally we derive golden-ratio-base BBP-type formulas for some mathematical constants, including $\pi$, $\log 2$, $\log \varphi$ and $\sqrt{2}\arctan \sqrt{2}$. The $\varphi -$nary BBP-type formulas derived here are considerably simpler than similar results contained in earlier literature.