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.