An Unusual Continued Fraction

We consider the real number $\sigma$ with continued fraction expansion $[a_0, a_1, a_2,\ldots] = [1,2,1,4,1,2,1,8,1,2,1,4,1,2,1,16,\ldots]$,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.