A066716 Decimal expansion of the binary Champernowne constant 0.862240125868... whose binary expansion is the concatenation of 1, 2, 3, ... written in binary.

8, 6, 2, 2, 4, 0, 1, 2, 5, 8, 6, 8, 0, 5, 4, 5, 7, 1, 5, 5, 7, 7, 9, 0, 2, 8, 3, 2, 4, 9, 3, 9, 4, 5, 7, 8, 5, 6, 5, 7, 6, 4, 7, 4, 2, 7, 6, 8, 2, 9, 9, 0, 9, 4, 5, 1, 6, 0, 7, 1, 2, 1, 4, 5, 5, 7, 3, 0, 6, 7, 4, 0, 5, 9, 0, 5, 1, 6, 4, 5, 8, 0, 4, 2, 0, 3, 8, 4, 4, 1, 4, 3, 8, 6, 1, 8, 1, 3, 3, 4

Offset

0,1

Comments

A theorem of Copeland & Erdős proves that this constant is 2-normal. - Charles R Greathouse IV, Feb 06 2015

This constant is transcendental. Note that this result is nontrivial: it is not a corollary of the result of Masaaki Amou saying that the base-b Champernowne constant has irrationality measure b, because the Thue-Siegel-Roth theorem only guarantees that a number with irrationality measure greater than 2 is transcendental. However, it is already stated in Masaaki Amou's paper that K. Mahler proved that the base-b Champernowne constant is transcendental for all b. - Jianing Song, Sep 27 2023

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Masaaki Amou, Approximation to certain transcendental decimal fractions by algebraic numbers, J. Number Theory, 37 (2) (1991), pp. 231-241.

A. H. Copeland and P. Erdős, Note on normal numbers, Bull. Amer. Math. Soc. 52 (1946), pp. 857-860.

Eric E. Weisstein, Binary Champernowne Constant.

Formula

The "binary" Champernowne constant is the number whose base-2 expansion is the concatenation of the binary representations of the integers, 0.(1)(10)(11)(100)(101)(110)(111)(1000)..., cf. A030302.

Example

0.8622401258680545715577902832493945785657647427682990945160712145573067405905...

Mathematica

a = {}; Do[a = Append[a, IntegerDigits[n, 2]], {n, 1, 100} ]; RealDigits[ N[ FromDigits[ {Flatten[a], 0}, 2], 100]]
First[RealDigits[ChampernowneNumber[2], 10, 100]] (* Paolo Xausa, Jun 12 2024 *)

Prog

(PARI) my(s=0.); forstep(n=default(realprecision), 1, -1, s=(s+n)>>#binary(n)); s \\ Charles R Greathouse IV, Feb 06 2015, corrected by M. F. Hasler, Mar 22 2017
(PARI) s=0; sum(n=1, 31, n*.5^s+=logint(n, 2)+1) \\ Accurate to 0.5^s. The sum up to n=31 is enough for standard precision of 38 digits. - M. F. Hasler, Mar 22 2017

Crossrefs

Cf. A030302 (binary digits), A030190 (same with initial 0), A030303 (indices of 1's), A007088, A047778 (concatenate binary 1..n).

Cf. A066717 (continued fraction), A365238 (reciprocal).

Cf. A100125 (Sum n/2^(n^2)).

Cf. A033307.

Sequence in context: A272430 A021120 A021541 * A339801 A360938 A272006

Adjacent sequences: A066713 A066714 A066715 * A066717 A066718 A066719

Keyword

cons,nonn,base,changed

Author

Robert G. Wilson v, Jan 14 2002

Extensions

Leading zero removed, offset adjusted, and keyword:cons added by R. J. Mathar, Mar 04 2010

Name edited by M. F. Hasler, Oct 26 2019

Status

approved