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

Kunle Adegoke, Jaume Oliver Lafont

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=α^n + β^…