Catalan's constant continued fraction is the simple continued fraction of Catalan's constant 
,
which begins ![[0;1,10,1,8,1,88,4,1,1,...]](/images/equations/CatalansConstantContinuedFraction/Inline2.svg)
(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 reported in 2013 are summarized below.
| terms | date | by |
 | Jul. 20, 2013 | E. Weisstein |
 | Aug. 7, 2013 | E. Weisstein |
Positions in the continued fraction are counted from 0, corresponding to the initial term 
. The plot above shows the positions of the first occurrences
of 1, 2, 3, ..., which begin 1, 13, 14, 7, 45, 36, 10, 4, 21, 2, ... (OEIS A196461).
The smallest positive integers reported not to
occur 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, 8708, 9075, ... (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 ![[a_0;a_1,a_2,...]](/images/equations/CatalansConstantContinuedFraction/Inline8.svg)
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).
