A033307 -id:A033307 - cp's OEIS frontend

A030998 Write n in base 7 and juxtapose.

0, 1, 2, 3, 4, 5, 6, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 4, 0, 4, 1, 4, 2, 4, 3, 4, 4, 4, 5, 4, 6, 5, 0, 5, 1, 5, 2, 5, 3, 5, 4, 5, 5, 5, 6, 6, 0, 6, 1, 6, 2, 6, 3, 6, 4, 6, 5, 6, 6

Offset: 0

Clark Kimberling

An irregular table in which the n-th row lists the base-7 digits of n. - Jason Kimberley, Dec 07 2012

The base-7 Champernowne constant: it is normal in base 7. - Jason Kimberley, Dec 07 2012

Tables in which the n-th row lists the base b digits of n: A030190 and A030302 (b=2), A003137 and A054635 (b=3), A030373 (b=4), A031219 (b=5), A030548 (b=6), this sequence (b=7), A031035 and A054634 (b=8), A031076 (b=9), A007376 and A033307 (b=10). - Jason Kimberley, Dec 06 2012

Cf. A007093.

[0]cat &cat[Reverse(IntegerToSequence(n,7)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

Flatten[IntegerDigits[#,7]&/@Range[0,60]]  (* Harvey P. Dale, Mar 04 2011 *)
almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; Array[ almostNatural[#, 7] &, 105, 0] (* Robert G. Wilson v, Jun 29 2014 *)

from itertools import count, chain, islice
from sympy.ntheory.factor_ import digits
def A030998_gen(): return chain.from_iterable(digits(m, 7)[1:] for m in count(0))
A030998_list = list(islice(A030998_gen(), 30)) # Chai Wah Wu, Jan 07 2022

A244677 The spiral of Champernowne, read along the East ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 04 2014

Inspired by Stanislaw Ulam's spiral, circa 1963.

			The beginning of the infinite spiral of David Gawen Champernowne:
.
  7--1--9--6--1--8--6--1--7--6--1--6--6--1--5--6--1--4--6--1--3  .
  |                                                           |  |
  0  1--4--4--1--3--4--1--2--4--1--1--4--1--0--4--1--9--3--1  6  .
  |  |                                                     |  |  |
  1  4  2--1--1--2--1--0--2--1--9--1--1--8--1--1--7--1--1  8  1  .
  |  |  |                                               |  |  |  |
  7  5  2  0--1--1--0--1--0--0--1--9--9--8--9--7--9--6  6  3  2  9
  |  |  |  |                                         |  |  |  |  |
  1  1  1  2  7--7--6--7--5--7--4--7--3--7--2--7--1  9  1  1  6  8
  |  |  |  |  |                                   |  |  |  |  |  |
  1  4  2  1  7  5--5--4--5--3--5--2--5--1--5--0  7  5  1  7  1  1
  |  |  |  |  |  |                             |  |  |  |  |  |  |
  7  6  3  0  8  5  7--3--6--3--5--3--4--3--3  5  0  9  5  3  1  8
  |  |  |  |  |  |  |                       |  |  |  |  |  |  |  |
  2  1  1  3  7  6  3  3--2--2--2--1--2--0  3  9  7  4  1  1  6  8
  |  |  |  |  |  |  |  |                 |  |  |  |  |  |  |  |  |
  1  4  2  1  9  5  8  2  3--1--2--1--1  2  2  4  9  9  1  6  1  1
  |  |  |  |  |  |  |  |  |           |  |  |  |  |  |  |  |  |  |
  7  7  4  0  8  7  3  4  1  5--4--3  1  9  3  8  6  3  4  3  0  7
  |  |  |  |  |  |  |  |  |  |     |  |  |  |  |  |  |  |  |  |  |
  3  1  1  4  0  5  9  2  4  6  1--2  0  1  1  4  8  9  1  1  6  8
  |  |  |  |  |  |  |  |  |  |        |  |  |  |  |  |  |  |  |  |
  1  4  2  1  8  8  4  5  1  7--8--9--1  8  3  7  6  2  1  5  1  1
  |  |  |  |  |  |  |  |  |              |  |  |  |  |  |  |  |  |
  7  8  5  0  1  5  0  2  5--1--6--1--7--1  0  4  7  9  3  3  9  6
  |  |  |  |  |  |  |  |                    |  |  |  |  |  |  |  |
  4  1  1  5  8  9  4  6--2--7--2--8--2--9--3  6  6  1  1  1  5  8
  |  |  |  |  |  |  |                          |  |  |  |  |  |  |
  1  4  2  1  2  6  1--4--2--4--3--4--4--4--5--4  6  9  1  4  1  1
  |  |  |  |  |  |                                |  |  |  |  |  |
  7  9  6  0  8  0--6--1--6--2--6--3--6--4--6--5--6  0  2  3  8  5
  |  |  |  |  |                                      |  |  |  |  |
  5  1  1  6  3--8--4--8--5--8--6--8--7--8--8--8--9--9  1  1  5  8
  |  |  |  |                                            |  |  |  |
  1  5  2  1--0--7--1--0--8--1--0--9--1--1--0--1--1--1--1  3  1  1
  |  |  |                                                  |  |  |
  7  0  7--1--2--8--1--2--9--1--3--0--1--3--1--1--3--2--1--3  7  4
  |  |                                                        |  |
  6  1--5--1--1--5--2--1--5--3--1--5--4--1--5--5--1--5--6--1--5  8
  |                                                              |
  1--7--7--1--7--8--1--7--9--1--8--0--1--8--1--1--8--2--1--8--3--1

Robert G. Wilson v, Cover of the March 1964 issue of Scientific American

Cf. A033307, A054552, A244678 - A244688, A033952, A244690 - A244692.

almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; f[n_] := 4n^2 - 11n + 8 (* see formula section *); Array[ almostNatural[ f@#, 10] &, 105]

Formulas for rays in directions of 32 compass points:
  SE     4n^2   -4n  +1
  SExS  64n^2 -113n +50
  SSE   16n^2  -25n +10
  SxE   64n^2 -115n +52
  S      4n^2   -5n  +2
  SxW   64n^2 -117n +54
  SSW   16n^2  -27n +12
  SWxS  64n^2 -119n +56
  SW     4n^2   -6n  +3
  SWxW  64n^2 -121n +58
  WSW   16n^2  -29n +14
  WxS   64n^2 -123n +60
  W      4n^2   -7n  +4
  WxN   64n^2 -125n +62
  WNW   16n^2  -31n +16
  NWxW  64n^2 -127n +64
  NW     4n^2   -8n  +5
  NWxN  64n^2 -129n +66
  NNW   16n^2  -33n +18
  NxW   64n^2 -131n +68
  N      4n^2   -9n  +6
  NxE   64n^2 -133n +70
  NNE   16n^2  -35n +20
  NExN  64n^2 -135n +72
  NE     4n^2  -10n  +7
  NExE  64n^2 -137n +74
  ENE   16n^2  -37n +22
  ExN   64n^2 -139n +76
  E      4n^2  -11n  +8
  ExS   64n^2 -141n +78
  ESE   16n^2  -39n +24
  SExE  64n^2 -143n +80

A031035 Write n in base 8 and juxtapose.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

Apart from the initial term, identical to A054634.
Should not be merged with A054634 because there are many sequences which depend on this sequence starting with a 1. - N. J. A. Sloane, Jan 30 2010
An irregular table in which the n-th row lists the base-8 digits of n. - Jason Kimberley, Dec 07 2012
The base-8 Champernowne constant: it is normal in base 8. - Jason Kimberley, Dec 07 2012

Robert Israel, Table of n, a(n) for n = 1..10000

Tables in which the n-th row lists the base b digits of n: A030190 and A030302 (b=2), A003137 and A054635 (b=3), A030373 (b=4), A031219 (b=5), A030548 (b=6), A030998 (b=7), this sequence and A054634 (b=8), A031076 (b=9), A007376 and A033307 (b=10). - Jason Kimberley, Dec 06 2012

Cf. A007094.

&cat[Reverse(IntegerToSequence(n,8)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

seq(op(ListTools:-Reverse(convert(n,base,8))),n=1..100); # Robert Israel, Nov 12 2024

Flatten[ IntegerDigits[ Range[40], 8]] (* or *)
almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; Array[ almostNatural[#, 8] &, 105] (* Robert G. Wilson v, Jun 29 2014 *)

from itertools import count, islice
from sympy.ntheory.factor_ import digits
def A031035_gen(): return (d for m in count(1) for d in digits(m,8)[1:])
A031035_list = list(islice(A031035_gen(),30)) # Chai Wah Wu, Jan 07 2022

A054634 Champernowne sequence: write n in base 8 and juxtapose.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Apr 16 2000

Apart from the initial term, identical to A031035.
Should not be merged with A031035 because there are many sequences which depend on the latter starting with a 1. - N. J. A. Sloane, Jan 30 2010
An irregular table in which the n-th row lists the base-8 digits of n. - Jason Kimberley, Dec 07 2012
The base-8 Champernowne constant: it is normal in base 8. - Jason Kimberley, Dec 07 2012

Cf. A007376, A030190, A007094.

Tables in which the n-th row lists the base b digits of n: A030190 and A030302 (b=2), A003137 and A054635 (b=3), A030373 (b=4), A031219 (b=5), A030548 (b=6), A030998 (b=7), A031035 and this sequence (b=8), A031076 (b=9), A007376 and A033307 (b=10). - Jason Kimberley, Dec 06 2012

[0]cat &cat[Reverse(IntegerToSequence(n,8)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

Flatten[ IntegerDigits[ Range[0, 40], 8]] (* or *)
almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; Array[ almostNatural[#, 8] &, 105, 0] (* Robert G. Wilson v, Jun 29 2014 *)

from itertools import count, chain, islice
from sympy.ntheory.factor_ import digits
def A054634_gen(): return chain.from_iterable(digits(m, 8)[1:] for m in count(0))
A054634_list = list(islice(A054634_gen(), 30)) # Chai Wah Wu, Jan 07 2022

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

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein

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

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

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

Cf. A033307, A033435.

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 *)

{ 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

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

A054635 Champernowne sequence: write n in base 3 and juxtapose.

Original entry on oeis.org

0, 1, 2, 1, 0, 1, 1, 1, 2, 2, 0, 2, 1, 2, 2, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 0, 1, 2, 1, 1, 2, 2, 2, 0, 0, 2, 0, 1, 2, 0, 2, 2, 1, 0, 2, 1, 1, 2, 1, 2, 2, 2, 0, 2, 2, 1, 2, 2, 2, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 2, 0, 1, 0, 2, 1

Offset: 0

N. J. A. Sloane, Apr 16 2000

Essentially the same as A003137. - R. J. Mathar, Aug 29 2009
An irregular table in which the n-th row lists the base-3 digits of n. - Jason Kimberley, Dec 07 2012
The base-3 Champernowne constant (A077771): it is normal in base 3. - Jason Kimberley, Dec 07 2012

Reinhard Zumkeller, Rows n = 0..1000 of triangle, flattened
Eric Weisstein's World of Mathematics, Ternary Champernowne Constant
Wikipedia, Ternary numeral system

Cf. A054637 (partial sums).
Cf. A081604 (row lengths), A053735 (row sums), A030341 (rows reversed), A007089, A077771.
Table in which the n-th row lists the base b digits of n: A030190 and A030302 (b=2), A003137 and this sequence (b=3), A030373 (b=4), A031219 (b=5), A030548 (b=6), A030998 (b=7), A031035 and A054634 (b=8), A031076 (b=9), A007376 and A033307 (b=10). - Jason Kimberley, Dec 06 2012

a054635 n k = a054635_tabf !! n !! k
a054635_row n = a054635_tabf !! n
a054635_tabf = map reverse a030341_tabf
a054635_list = concat a054635_tabf
-- Reinhard Zumkeller, Feb 21 2013

[0]cat &cat[Reverse(IntegerToSequence(n,3)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; Array[ almostNatural[#, 3] &, 105, 0] (* Robert G. Wilson v, Jun 29 2014 *)
First[RealDigits[ChampernowneNumber[3], 3, 100, 0]] (* Paolo Xausa, Jun 19 2024 *)

from sympy.ntheory.digits import digits
def agen(limit):
    for n in range(limit):
        yield from digits(n, 3)[1:]
print([an for an in agen(35)]) # Michael S. Branicky, Sep 01 2021

A066717 The continued fraction for the "binary" Champernowne constant.

Original entry on oeis.org

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, 1, 4, 5, 1, 2, 1, 7, 1, 16, 1, 4, 1, 5, 5, 1, 5, 1, 4, 1, 2, 1, 5, 3, 2, 38, 2, 12, 1, 15, 2, 6, 3, 30, 4682854730443938, 1, 1, 68, 1, 6, 5, 4, 4, 1, 2, 1, 1, 1, 1, 2, 22, 1, 2, 7, 1, 2

Offset: 0

Robert G. Wilson v, Jan 14 2002

Robert G. Wilson v, Table of n, a(n) for n = 0..1000
J. K. Sikora, The first 98093504 CFE coefficients of the binary Champernowne Constant (231 MB zipped)
Eric E. Weisstein, Binary Champernowne Constant

Cf. A030190 & A066716 (binary & decimal digits of the binary Champernowne constant), A033307 (decimal Champernowne constant).

Cf. A054635, A077771, A077772: base 3, decimals and continued fraction of ternary Champernowne constant.

a = {}; Do[a = Append[a, IntegerDigits[n, 2]], {n, 1, 10^3} ]; ContinuedFraction[ N[ FromDigits[ {Flatten[a], 0}, 2], 500]]
almostNatural[n_, b_] :=  Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; Take[ ContinuedFraction[ FromDigits[ {Array[almostNatural[#, 2] &, 20000], 0}, 2]], 100] (* Robert G. Wilson v, Jul 21 2014 *)

A066717(b=2,t=1.,s=b)={contfrac(sum(n=1,default(realprecision)*2.303\log(b)+1, nM. F. Hasler, Oct 25 2019

A077772 Continued fraction expansion of the ternary Champernowne constant.

Original entry on oeis.org

0, 1, 1, 2, 37, 1, 162, 1, 1, 1, 3, 1, 7, 1, 9, 2, 3, 1, 3068518062211324, 2, 1, 2, 6, 13, 1, 2, 1, 3, 1, 10, 1, 21, 1, 1, 4, 3, 577, 1, 1079268324684171943515797470873767312825026176345571319042096689270, 1, 1, 1, 3, 4, 21, 3, 1, 9, 1

Offset: 0

Eric W. Weisstein, Nov 15 2002

John K. Sikora, Table of n, a(n) for n = 0..2061 (terms n = 0..1155 from Robert G. Wilson v)
J. K. Sikora, The first 2982556 terms of the CFE of the ternary Champernowne Constant (141 MB zipped)
Eric Weisstein's World of Mathematics, Ternary Champernowne Constant

Cf. A054635 (ternary digits), A077771 (decimals).

Cf. A030190, A066716, A066717: binary digits, decimals and continued fraction of the binary Champernowne constant; A033307: decimal Champernowne constant.

almostNatural[n_, b_] :=  Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; Take[ ContinuedFraction[ FromDigits[ {Array[almostNatural[#, 3] &, 20000], 0}, 3]], 100] (* Robert G. Wilson v, Jul 21 2014 *)

\p 10000
t=0;r=0.;T=1; for(n=1,1e6,d=#digits(n,3);t+=d;T*=3^d;r+=n/T;if(t>20959, return)); v=contfrac(r); v[1..30] \\ Charles R Greathouse IV, Sep 23 2014

A077772(b=3,t=1.,s=b)={contfrac(sum(n=1,default(realprecision)*2.303/log(b)+1, nM. F. Hasler, Oct 25 2019

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

Original entry on oeis.org

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

Robert G. Wilson v, Jan 14 2002

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

			0.8622401258680545715577902832493945785657647427682990945160712145573067405905...

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.

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.

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 *)

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

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

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.

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

A033988 Write 0,1,2,... in a clockwise spiral on a square lattice, writing each digit at a separate lattice point, starting with 0 at the origin and 1 at x=0, y=-1; sequence gives the numbers on the positive y-axis.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

In other words, write 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 ... in a clockwise spiral, starting with the 0 and taking the first step south; the sequence is then picked out from the resulting spiral by starting at the origin and moving north.

			  1---3---1---4---1
  |               |
  2   4---5---6   5
  |   |       |   |
  1   3   0   7   1
  |   |   |   |   |
  1   2---1   8   6
  |           |   |
  1---0---1---9   1
.
We begin with the 0 and wrap the numbers 1 2 3 4 ... around it.
Then the sequence is obtained by reading vertically upwards, starting from the initial 0.

Andrew Woods, Table of n, a(n) for n = 0..1000

Sequences based on the same spiral: A033953, A033989, A033990. Spiral without zero: A033952.
Other sequences from spirals: A001107, A002939, A007742, A033951, A033954, A033991, A002943, A033996.
Cf. A033307.

nmax = 105; A033307 = Flatten[IntegerDigits /@ Range[0, nmax^2 + 10 nmax]]; a[n_] := If[n==0, 0, A033307[[4n^2 + n + 1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Apr 24 2017, after Andrew Woods *)

a(n) = A033307(4*n^2 + n - 1) for n > 0. - Andrew Woods, May 18 2012

More terms from Andrew Gacek (andrew(AT)dgi.net)

Edited by Jon E. Schoenfield, Aug 12 2018

cp's OEIS Frontend

A030998 Write n in base 7 and juxtapose.

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A244677 The spiral of Champernowne, read along the East ray.

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A031035 Write n in base 8 and juxtapose.

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A054634 Champernowne sequence: write n in base 8 and juxtapose.

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

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A054635 Champernowne sequence: write n in base 3 and juxtapose.

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A066717 The continued fraction for the "binary" Champernowne constant.

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A077772 Continued fraction expansion of the ternary Champernowne constant.

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