An Unusual Continued Fraction

We consider the real number $σ$ 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 $σ^2$ and demonstrate that $σ^2$ (and $σ$) are both transcendental numbers. We also show that certain partial quotients of $σ^2$ grow doubly exponentially, thus confirming a conjecture of Hanna and Wilson.

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