The "binary" Champernowne constant is obtained by concatenating the binary representations of the integers
 |  |  |
(1)
|
 |  |  |
(2)
|
(OEIS A030190 and A066716). The sequence given by the first few concatenations is therefore 1, 110, 11011, 11011100,
11011100101, ... (OEIS A058935). 
can also be written
 |
(3)
|
with
 |
(4)
|
and 
the floor function (Bailey and Crandall 2002).
Interestingly, 
is 2-normal (Bailey and
Crandall 2002).
has continued fraction [0, 1, 6, 3, 1, 6, 5,
3, 3, 1, 6, 4, 1, 3, 298, 1, 6, 1, 1, 3, 285, 7, 2, 4, 1, 2, 1, 2, 1, 1, 4534532,
...] (OEIS A066717), which exhibits sporadic
large terms. The numbers of decimal digits in these terms are 0, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 3, ....
