A030167 Continued fraction expansion of the Champernowne constant 0.1234567891011121314...

0, 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn
• cofr
• nice

Comments

The next term, a(18) = 457540111...783010987 has 166 digits.

It is followed by a(19 .. 39) = (6, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2, 1, 83, 1, 156, 4, 58, 8, 54). - M. F. Hasler, Oct 25 2019

a(40) = 445735380...113172423 has 2504 digits. - Harvey P. Dale, May 23 2015, index corrected by M. F. Hasler, Oct 25 2019

Examples

			This is the continued fraction of the number 0.123456789101112131415... whose decimals are obtained by concatenating the base-10 representations of all positive integers.
		

Links

• Harry J. Smith, Table of n, a(n) for n=0..39 (see the a-file for further terms)
• Harry J. Smith, Table of n, a(n) for n=0..161
• H. Havermann, 13522 (less the 3 largest) terms (2 MB file)
• J. K. Sikora, Champernowne's Constant CFE Coefficients to HWM #12 (789 MB unzipped)
• Eric Weisstein's World of Mathematics, Champernowne Constant.
• G. Xiao, Contfrac
• Index entries for continued fractions for constants

Crossrefs

Cf. A033307, A033435.

Programs

Mathematica

f[0] = 0; f[n_Integer] := 10^(Floor[Log[10, n]] + 1)*f[n - 1] + n; ContinuedFraction[ N[ f[211]/ 10^(Floor[ Log[10, f[211] ]] + 1), Floor[ Log[10, f[211] ]] + 1], 19 ]
chcon=Module[{con=FromDigits[Flatten[IntegerDigits/@Range[250]]]}, N[con/10^IntegerLength[con],IntegerLength[con]]]; ContinuedFraction[ chcon,19] (* Harvey P. Dale, Sep 18 2011 *)
ContinuedFraction[N[ChampernowneNumber[10],10000]] (* Harvey P. Dale, May 23 2015 *)

PARI

{ default(realprecision, 6000); x=0; y=1; d=10.0; e=1.0; n=0; while (y!=x, y=x; n++; if (n==d, d=d*10); e=e*d; x=x+n/e; ); x=contfrac(x); for (n=1, 160, write("b030167.txt", n-1, " ", x[n])); write("b030167.txt", "160 1"); write("b030167.txt", "161 1"); } \\ Harry J. Smith, Apr 18 2009

Extensions

Edited by Daniel Forgues, Apr 01 2010, M. F. Hasler, Oct 25 2019