I've been looking into Apéry's irrationality proof of $\zeta (3)$, and one of the first questions I instantly had, was how did he derive the following continued fraction? $$\begin{equation*} \zeta (3)=\dfrac{6}{5+\overset{\infty }{\underset{n=1}{\mathbb{K}}}\dfrac{-n^{6}}{34n^{3}+51n^{2}+27n+5}}\end{equation*}$$

Furthermore, is it possible to get a similar continued fraction for $\zeta(5)$, $\zeta(7)$ or $G$?

A rapidly converging central binomial series was recently found for Catalan's constant:

$$G = \frac{1}{2} \sum_{n=0}^{\infty} (-1)^n \frac{(3n+2) 8^n}{(2n+1)^3 \binom{2n}{n}^3}$$ If I understand correctly, this gives us the first stage of an analogue of Apéry's proof for $G$. The next stage in his proof is to use a fast recursion formula that approaches $\zeta(3)$: $$n^3 u_n + (n-1)^3 u_{n-2} = (34n^3 - 51n^2 + 27n - 5) u_{n-1},\, n \geq 2$$ Can a similar recursive formula be found for $G$?