Soldner's Constant Continued Fraction

The continued fraction for  is given by [1; 2, 4, 1, 1, 1, 3, 1, 1, 1, 2, 47, 2, ...] (OEIS A099803).

The positions at which the numbers 1, 2, ... occur in the continued fraction are 0, 1, 6, 2, 47, 28, 21, 107, 114, ... (OEIS A000000).

The high-water marks are 1, 2, 4, 47, 99, 294, 527, 616, 1152, ... (OEIS A099804), which occur at positions 0, 1, 2, 11, 69, 125, 201, 584, 1591, 2435, ... (OEIS A229230).

Let the continued fraction of  be denoted  and let the denominators of the convergents be denoted , , ..., . Then plots above show successive values of , , , which appear to converge to Khinchin's constant (left figure) and , which appear to converge to the Lévy constant (right figure), although neither of these limits has been rigorously established.

REFERENCES:

Sloane, N. J. A. Sequences A099803, A099804, and A229230 in "The On-Line Encyclopedia of Integer Sequences."