The Glaisher-Kinkelin constant continued fraction is the simple continued fraction of the Glaisher-Kinkelin
constant 
, which begins ![[1;3,1,1,5,1,1,1,3,12,4,1,271,1,...]](/images/equations/Glaisher-KinkelinConstantContinuedFraction/Inline2.svg)
(OEIS A087501).
A plot of the first 256 terms of the continued
fraction represented as a sequence of binary bits is shown above.
Write the continued fraction as ![[a_0;a_1,a_2,...]](/images/equations/Glaisher-KinkelinConstantContinuedFraction/Inline3.svg)
, where 
. The first occurrences of the terms 1, 2, 3, ...
have indices 
, 15, 1, 10, 4, 19, 16, 77, 21, 62, 229, 9, 52, ...
(OEIS A225762). The smallest term whose first
occurrence was unresolved in a 2013 computation was 204, with 
(E. Weisstein, Jul. 25, 2013).
The consecutively largest terms are 1, 3, 5, 12, 271, 12574, 13740, 78907, 133430, 574536, ... (OEIS A099791), occurring at positions 0, 1, 4, 9, 12, 266, 3170, 3212, 12961, 82527, ... (OEIS A225752).
Let 
be the denominator of the 
th convergent. The 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.
