An Unusual Continued Fraction

Abstract

We consider the real number $\sigma$ with continued fraction expansion $[a_0,a_1,a_2,\dots ]=[1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,\dots ]$, where $a_i$ is the largest power of $2$ dividing $i+1$. We compute the irrationality measure of ${\sigma }^2$ and demonstrate that ${\sigma }^2$ (and $\sigma$) are both transcendental numbers. We also show that certain partial quotients of ${\sigma }^2$ grow doubly exponentially, thus confirming a conjecture of Hanna and Wilson.

12 pages