Golden Ratio Base Expansions of the Logarithm and Inverse Tangent of Fibonacci and Lucas Numbers

Abstract

Let $\alpha =(1+\sqrt{5})/2$, the golden ratio, and $\beta =-1/\alpha =(1-\sqrt{5})/2$. Let $F_n$ and $L_n$ be the Fibonacci and Lucas numbers, defined by $F_n=({\alpha }^n-{\beta }^n)/\sqrt{5}$ and $L_n={\alpha }^n+{\beta }^n$, for all non-negative integers. We derive base~$\alpha$ expansions of $\log F_n$, $\log L_n$, $\arctan \frac{1}{F_n}$ and $\arctan \frac{1}{L_n}$ for all positive integers $n$.

18 pages, no figures or tables