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, ....

REFERENCES:

Sloane, N. J. A. Sequences A030190, A066716, and A066717 in "The On-Line Encyclopedia of Integer Sequences."