Transcendental number(s) with all positive integers in order
In mathematics, the Champernowne constantC10 is a transcendental real constant whose decimal expansion has important properties. It is named after economist and mathematician D. G. Champernowne, who published it as an undergraduate in 1933.[1] The number is defined by concatenating the base 10 representations of the positive integers:
C10 = 0.12345678910111213141516... (sequence A033307 in the OEIS).
Champernowne constants can also be constructed in other bases similarly; for example,
C2 = 0.11011100101110111... 2
and
C3 = 0.12101112202122... 3.
The Champernowne word or Barbier word is the sequence of digits of C10 obtained by writing it in base 10 and juxtaposing the digits:[2][3]
12345678910111213141516... (sequence A007376 in the OEIS)
More generally, a Champernowne sequence (sometimes also called a Champernowne word) is any sequence of digits obtained by concatenating all finite digit-strings (in any given base) in some recursive order.[4]
For instance, the binary Champernowne sequence in shortlex order is
0 1 00 01 10 11 000 001 ... (sequence A076478 in the OEIS)
where spaces (otherwise to be ignored) have been inserted just to show the strings being concatenated.
^Champernowne 1933
^Cassaigne & Nicolas (2010) p.165
^Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. p. 299. ISBN 978-0-521-82332-6. Zbl 1086.11015.
^
Calude, C.; Priese, L.; Staiger, L. (1997), Disjunctive sequences: An overview, University of Auckland, New Zealand, pp. 1–35, CiteSeerX 10.1.1.34.1370
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