Improved continued fraction sequence convergent to the Somos' quadratic recurrence constant (Q905979)

The sequence \(g_n=ng^2_{n-1}\) with \(g_0=1\) (defined by Somos) satisfies the asymptotic formula \[ g_n=\sigma^{2^n}\left(n+2-\frac{1}{n}+\frac{4}{n^2}-\frac{21}{n^3}+\cdots\right)^{-1} \] where \(\sigma\) is now known as the Somos' quadratic recurrence constant. Using the fact that \(\sigma=2\exp\{-\frac{1}{2}\gamma(\frac{1}{2})\}\), the authors present an approximation for \(\sigma\) by approximating \(\gamma(\frac{1}{2})\), using the generalized Euler constant function \[ \gamma(z)=\sum\limits_{k=1}^\infty z^{k-1}\left(\frac{1}{k}-\ln\frac{k+1}{k}\right)\quad (|z|\leq 1). \] They also present an approximation for the constant \(\gamma(\frac{1}{3})\).