Champernowne Constant
Champernowne's number 0.1234567891011... (Sloane's A033307) is the decimal obtained by concatenating the PositiveIntegers. It is Normal in base 10. In 1961, Mahler showed it to also beTranscendental.
The Continued Fraction of the Champernowne constant is [0, 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15,
45754011139103107648364662824295611859960393971045755500066200439309026265925631493795320774712865631386412093755035520946071830899845758014698631488335921417830109876, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2, 1, 83, 1, 156, 4, 58, 8, 54, ...] (Sloane's A030167). The next term of theContinued Fraction is huge, having 2504 digits. In fact, the coefficients eventually become unbounded, making thecontinued fraction difficult to calculate for too many more terms. 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 (Plouffe). Interestingly, the Copeland-Erdös Constant, which is the decimal obtained byconcatenating the Primes, has a well-behaved Continued Fraction which does not show the ``large term'' phenomenon.
References
Champernowne, D. G. ``The Construction of Decimals Normal in the Scale of Ten.'' J. London Math. Soc. 8, 1933. Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/cntfrc/cntfrc.html Sloane, N. J. A.A030167 andA033307 in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.
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