(noch nicht übersetzt)

An irrational number $x$ can be uniquely expressed as a continued fraction $[a_0; a_1,a_2,a_3,\dots]$: $$ x=a_{0}+\cfrac{1}{a_1+\cfrac{1}{a_2+\cfrac{1}{a_3+{_\ddots}}}} $$where $a_0$ is an integer and $a_1,a_2,a_3,\dots$ are positive integers.

Define $k_j(x)$ to be the geometric mean of $a_1,a_2,\dots,a_j$.
That is, $k_j(x)=(a_1a_2 \cdots a_j)^{1/j}$.
Also define $k_\infty(x)=\lim_{j\to \infty} k_j(x)$.

Khinchin proved that almost all irrational numbers $x$ have the same value of $k_\infty(x)\approx2.685452\dots$ known as Khinchin's constant. However, there are some exceptions to this rule.

For $n\geq 0$ define $$\rho_n = \sum_{i=0}^{\infty} \frac{2^n}{2^{2^i}} $$For example $\rho_2$, with continued fraction beginning $[3; 3, 1, 3, 4, 3, 1, 3,\dots]$, has $k_\infty(\rho_2)\approx2.059767$.

Find the geometric mean of $k_{\infty}(\rho_n)$ for $0\leq n\leq 50$, giving your answer rounded to six digits after the decimal point.