The first few terms in the continued fraction of the Champernowne constant are [0; 8,
9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 45754...10987, 6, 1, 1, 21,
...] (OEIS A030167), and the number of decimal
digits in these terms are 0, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 166,
1, ... (OEIS A143532). E. W. Weisstein
computed 
terms of the continued fraction on Jun. 30, 2013
using the Wolfram Language.
First occurrences of the terms 1, 2, 3, ... in the continued fraction ![[0;a_1,a_2,...,a_n]](/images/equations/ChampernowneConstantContinuedFraction/Inline2.svg)
occur at 
, 28, 13, 9, 93, 20, 31, 2, 3, 339, 71, 126, 107, ... (OEIS
A038706). The smallest unknown value is 188,
which has 
.
The continued fraction contains sporadic very large terms, making the continued fraction difficult to calculate. However, the size of the continued fraction high-water marks
display apparent patterns (Sikora 2012). Large terms greater than 
occur at positions 5, 19, 41, 102, 163, 247, 358, 460,
... and have 6, 166, 2504, 140, 33102, 109, 2468, 136, ... digits, respectively.
The high-water marks in terms of the continued fraction occur for terms 0, 1, 2, 4, 18, 40, 162, 526, 1708, 4838, 13522, 34062,
... (OEIS A143533; Sikora 2012), which have
0, 1, 1, 6, 166, 2504, 33102, 411100, 4911098, 57111096, 651111094, 7311111092, ...
(OEIS A143534; Sikora 2012) decimal digits,
respectively. Sikora (2012) conjectured that the number of decimal digits in the
th
high-water mark for 
are given by
 |
(1)
|
where
 |  |  |
(2)
|
 |  |  |
(3)
|
in agreement with known calculated values up to 
.
Rytin, M. "Champernowne Constant and Its Continued Fraction Expansion."