Independent confirmation of the second derivative C''(2)

Confirm independently that the second derivative of the asymptotic constant C(x_0)=\lim_{n\to\infty}(x_n-n) for the recurrence x_{n+1}=x_n+1+1/x_n^2, evaluated at x_0=2, is C''(2)\approx0.37462642198301734111.

Background

For the recurrence x_{n+1}=x_n+1+1/x_n^2, the paper defines C(x_0)=\lim_{n\to\infty}(x_n-n) and computes numerical estimates for C(2), C'(2), and C''(2). An initially derived series-based estimate for C''(2) is approximately 0.3909, but the authors identify a possible failure of the required uniform convergence assumption.

Using a high-order asymptotic expansion and centered finite differences with \varepsilon=10^{-20}, the paper obtains C''(2)\approx0.37462642198301734111. The discrepancy between these values is noted as unresolved, and the paper calls for an independent confirmation of the latter estimate.