The continued fraction for Apéry's constant  is [1; 4, 1, 18, 1, 1, 1, 4, 1, ...] (OEIS A013631).

The positions at which the numbers 1, 2, ... occur in the continued fraction are 0, 11, 24, 1, 63, 26, 16, 139, 9, 118, 20, ... (OEIS A229057). The incrementally maximal terms are 1, 4, 18, 30, 428, 458, 527, ... (OEIS A033166), which occur at positions 0, 1, 3, 28, 62, 571, 1555, 2012, 2529, ... (OEIS A229055).

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 A013631, A033166, A229055, and A229057 in "The On-Line Encyclopedia of Integer Sequences."