Catalan's Constant Continued Fraction

The simple continued fraction representations for Catalan's constant  is [0, 1, 10, 1, 8, 1, 88, 4, 1, 1, ...] (OEIS A014538). A plot of the first 256 terms of the continued fraction represented as a sequence of binary bits is shown above.

Record computations are summarized below.

terms	date	by
	Jul. 20, 2013	E. Weisstein
	Aug. 7, 2013	E. Weisstein

The plot above shows the positions of the first occurrences of 1, 2, 3, ... in the continued fraction, the first few of which are 1, 13, 14, 7, 45, 36, 10, 4, 21, 2, ... (OEIS A196461; illustrated above). The smallest number not occurring in the first  terms of the continued fraction are 31516, 31591, 32600, 32806, 33410, ... (E. Weisstein, Aug. 8, 2013).

The cumulative largest terms in the continued fraction are 0, 1, 10, 88, 322, 330, 1102, 6328, ... (OEIS A099789), which occur at positions 0, 1, 2, 6, 105, 284, 747, 984, 2230, 5377, ... (OEIS A099790).

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.

The Engel expansion of  is given by 2, 2, 2, 4, 4, 5, 5, 12, 13, 41, 110, ... (OEIS A054543).

REFERENCES:

Sloane, N. J. A. Sequences A014538, A054543, A099789, A099790, and A196461 in "The On-Line Encyclopedia of Integer Sequences."