A030998 -id:A030998 - cp's OEIS frontend

A030190 Binary Champernowne sequence (or word): write the numbers 0,1,2,3,4,... in base 2 and juxtapose.

0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0

Offset: 0

Clark Kimberling

a(A003607(n)) = 0 and for n > 0: a(A030303(n)) = 1. - Reinhard Zumkeller, Dec 11 2011
An irregular table in which the n-th row lists the bits of n (see the example section). - Jason Kimberley, Dec 07 2012
The binary Champernowne constant: it is normal in base 2. - Jason Kimberley, Dec 07 2012
This is the characteristic function of A030303, which gives the indices of 1's in this sequence and has first differences given by A066099. - M. F. Hasler, Oct 12 2020

			As an array, this begins:
0,
1,
1, 0,
1, 1,
1, 0, 0,
1, 0, 1,
1, 1, 0,
1, 1, 1,
1, 0, 0, 0,
1, 0, 0, 1,
1, 0, 1, 0,
1, 0, 1, 1,
1, 1, 0, 0,
1, 1, 0, 1,
1, 1, 1, 0,
1, 1, 1, 1,
1, 0, 0, 0, 0,
1, 0, 0, 0, 1,
...

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jean Berstel, Home Page (in case the following link should be broken)
Jean Berstel and Juhani Karhumäki, Combinatorics on words-a tutorial. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, # 79, pp. 178-228, 2003.
S. Ferenczi, Complexity of sequences and dynamical systems, Discrete Math., 206 (1999), 145-154.
Eric Weisstein's World of Mathematics, Binary Champernowne Constant
Index entries for transcendental numbers.

Cf. A007376, A003137, A030308. Same as and more fundamental than A030302, but I have left A030302 in the OEIS because there are several sequences that are based on it (A030303 etc.). - N. J. A. Sloane.
a(n) = T(A030530(n), A083652(A030530(n))-n-1), T as defined in A083651, a(A083652(k))=1.
Tables in which the n-th row lists the base b digits of n: this sequence and A030302 (b=2), A003137 and A054635 (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
A076478 is a similar sequence.
For run lengths see A056062; see also A318924.
See also A066099 for (run lengths of 0s) + 1 = first difference of positions of 1s given by A030303.

import Data.List (unfoldr)
a030190 n = a030190_list !! n
a030190_list = concatMap reverse a030308_tabf
-- Reinhard Zumkeller, Jun 16 2012, Dec 11 2011

[0]cat &cat[Reverse(IntegerToSequence(n,2)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

Flatten[ Table[ IntegerDigits[n, 2], {n, 0, 26}]] (* Robert G. Wilson v, Mar 08 2005 *)
First[RealDigits[ChampernowneNumber[2], 2, 100, 0]] (* Paolo Xausa, Jun 16 2024 *)

A030190_row(n)=if(n,binary(n),[0]) \\ M. F. Hasler, Oct 12 2020

from itertools import count, islice
def A030190_gen(): return (int(d) for m in count(0) for d in bin(m)[2:])
A030190_list = list(islice(A030190_gen(),30)) # Chai Wah Wu, Jan 07 2022

A030302 Write n in base 2 and juxtapose; irregular table in which row n lists the binary expansion of n.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1

Offset: 1

Clark Kimberling

The binary Champernowne constant: it is normal in base 2. - Jason Kimberley, Dec 07 2012
A word that is recurrent, but neither morphic nor uniformly recurrent. - N. J. A. Sloane, Jul 14 2018
See A030303 for the indices of 1's (so this is the characteristic function of A030303), with first differences (i.e., run lengths of 0's, increased by 1, with two consecutive 1's delimiting a run of zero 0's) given by A066099. - M. F. Hasler, Oct 12 2020

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

R. J. Mathar, Table of n, a(n) for n = 1..3998
Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit and Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017.

Essentially the same as A007088 and A030190. Cf. A030303, A007088.
Tables in which the n-th row lists the base b digits of n: A030190 and this sequence (b=2), A003137 and A054635 (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]
Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.

&cat[Reverse(IntegerToSequence(n,2)): n in [1..31]]; // Jason Kimberley, Mar 02 2012

A030302 := proc(n) local i,t1,t2; t1:=convert(n,base,2); t2:=nops(t1); [seq(t1[t2+1-i],i=1..t2)]; end; # N. J. A. Sloane, Apr 08 2021

i[n_] := Ceiling[FullSimplify[ProductLog[Log[2]/2 (n - 1)]/Log[2] + 1]]; a[n_] := Mod[Floor[2^(Mod[n + 2^i[n] - 2, i[n]] - i[n] + 1) Ceiling[(n + 2^i[n] - 1)/i[n] - 1]], 2]; (* David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 19 2007 *)
Join @@ Table[ IntegerDigits[i, 2], {i, 1, 40}] (* Olivier Gérard, Mar 28 2011 *)
Flatten@ IntegerDigits[ Range@ 25, 2] (* 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[#, 2] &, 105] (* Robert G. Wilson v, Jun 29 2014 *)

from itertools import count, islice
def A030302_gen(): # generator of terms
    return (int(d) for n in count(1) for d in bin(n)[2:])
A030302_list = list(islice(A030302_gen(),30)) # Chai Wah Wu, Feb 18 2022

a(n) = (floor(2^(((n + 2^i - 2) mod i) - i + 1) * ceiling((n + 2^i - 1)/i - 1))) mod 2 where i = ceiling( W(log(2)/2 (n - 1))/log(2) + 1 ) and W denotes the principal branch of the Lambert W function. See also Mathematica code. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 19 2007

A030548 Write n in base 6 and juxtapose.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

Base-6 analog of what in base 7 is A030998, in base 10 is A007376. In general, the Barbier infinite word base n (in this case, 6). - Jonathan Vos Post, May 13 2007
An irregular table in which the n-th row lists the base-6 digits of n. - Jason Kimberley, Dec 07 2012
The base-6 Champernowne constant: It is normal in base 6. - Jason Kimberley, Dec 07 2012

R. J. Mathar, Table of n, a(n) for n = 1..5950

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), this sequence (b=6), A030998 (b=7), A031035 and A054634 (b=8), A031076 (b=9), A007376 and A033307 (b=10). - Jason Kimberley, Dec 06 2012

Cf. A030567 for the same table with reversed rows.

&cat[Reverse(IntegerToSequence(n,6)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

Flatten@ IntegerDigits[ Range@ 50, 6] (* 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[ a[#, 6] &, 105] (* Robert G. Wilson v, Jul 01 2014 *)

from itertools import count, chain, islice
from sympy.ntheory.factor_ import digits
def A030548_gen(): return chain.from_iterable(digits(m, 6)[1:] for m in count(1))
A030548_list = list(islice(A030548_gen(), 30)) # Chai Wah Wu, Jan 07 2022

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Aug 23 2007

A031219 Write n in base 5 and juxtapose.

Original entry on oeis.org

1, 2, 3, 4, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 4, 0, 4, 1, 4, 2, 4, 3, 4, 4, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 3, 1, 0, 4, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 4, 1, 2, 0, 1, 2, 1, 1, 2, 2, 1, 2, 3, 1, 2, 4, 1

Offset: 1

Clark Kimberling

An irregular table in which the n-th row lists the base-5 digits of n. - Jason Kimberley, Dec 07 2012

The base-5 Champernowne constant: it is normal in base 5. - 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), this sequence (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

&cat[Reverse(IntegerToSequence(n,5)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

Flatten@ IntegerDigits[ Range@ 40, 5] (* 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[ a[#, 5] &, 105] (* Robert G. Wilson v, Jul 01 2014 *)

from itertools import count, chain, islice
from sympy.ntheory.factor_ import digits
def A031219_gen(): return chain.from_iterable(digits(m, 5)[1:] for m in count(1))
A031219_list = list(islice(A031219_gen(), 30)) # Chai Wah Wu, Jan 07 2022

A003137 Write n in base 3 and juxtapose.

Original entry on oeis.org

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, 1, 0, 2, 2, 1

Offset: 1

N. J. A. Sloane

An irregular table in which the n-th row lists the base-3 digits of n, see A007089. - Jason Kimberley, Dec 07 2012

The base-3 Champernowne constant (A077771): it is normal in base 3. - Jason Kimberley, Dec 07 2012

			1,
2,
1,0,
1,1,
1,2,
2,0,
2,1,
2,2,
1,0,0,
1,0,1,.... _R. J. Mathar_, Aug 16 2021

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system

Tables in which the n-th row lists the base b digits of n: A030190 and A030302 (b=2), this sequence and A054635 (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

Cf. A081604 (row lengths), A053735 (row sums), A030341 (rows reversed), A077771, A007089.

a003137 n k = a003137_tabf !! (n-1) !! k
a003137_row n = a003137_tabf !! (n-1)
a003137_tabf = map reverse $ tail a030341_tabf
a003137_list = concat a003137_tabf
-- Reinhard Zumkeller, Feb 21 2013

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

Flatten@ IntegerDigits[ Range@ 40, 3] (* 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[ a[#, 3] &, 105] (* Robert G. Wilson v, Jul 01 2014 *)

from itertools import count, islice
from sympy.ntheory.factor_ import digits
def A003137_gen(): return (d for m in count(1) for d in digits(m,3)[1:])
A003137_list = list(islice(A003137_gen(),30)) # Chai Wah Wu, Jan 07 2022

More terms from Larry Reeves (larryr(AT)acm.org), Sep 25 2000

A030373 Write n in base 4 and juxtapose.

Original entry on oeis.org

1, 2, 3, 1, 0, 1, 1, 1, 2, 1, 3, 2, 0, 2, 1, 2, 2, 2, 3, 3, 0, 3, 1, 3, 2, 3, 3, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 3, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 0, 1, 2, 1, 1, 2, 2, 1, 2, 3, 1, 3, 0, 1, 3, 1, 1, 3, 2, 1, 3, 3, 2, 0, 0, 2, 0, 1, 2, 0, 2, 2, 0, 3, 2, 1, 0

Offset: 1

Clark Kimberling

An irregular table in which the n-th row lists the base-4 digits of n. - Jason Kimberley, Nov 26 2012

The base-4 Champernowne constant: it is normal in base 4. - Jason Kimberley, Nov 26 2012

			1;
2;
3;
1,0;
1,1;
1,2;
1,3;
2,0;
2,1;
2,2;
...
3,3;
1,0,0;
1,0,1;
1,0,2;
1,0,3;
1,1,0; ....

R. J. Mathar, Table of n, a(n) for n = 1..4664
F. J. Aragon Artacho, D. H. Bailey, J. M. Borwein, and P. B. Borwein, Walking on real numbers, preprint September 2012.

Cf. A007376, A003137, A007090.

Tables in which the n-th row lists the base b digits of n: A030190 and A030302 (b=2), A003137 and A054635 (b=3), this sequence (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

&cat[Reverse(IntegerToSequence(n,4)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

Flatten[IntegerDigits[Range[40],4]] (* Harvey P. Dale, Aug 23 2011 *)

from itertools import count, chain, islice
from sympy.ntheory.factor_ import digits
def A030373_gen(): return chain.from_iterable(digits(m, 4)[1:] for m in count(1))
A030373_list = list(islice(A030373_gen(), 30)) # Chai Wah Wu, Jan 07 2022

A031076 Write n in base 9 and juxtapose.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

An irregular table in which the n-th row lists the base-9 digits of n. - Jason Kimberley, Dec 07 2012

The base-9 Champernowne constant: it is normal in base 9. - Jason Kimberley, Dec 07 2012

Reinhard Zumkeller, 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), A031035 and A054634 (b=8), this sequence (b=9), A007376 and A033307 (b=10). - Jason Kimberley, Dec 06 2012

Cf. A031087, A007095.

a031076 n = a031076_list !! (n-1)
a031076_list = concat $ map reverse $ tail a031087_tabf
-- Reinhard Zumkeller, Jul 07 2015

&cat[Reverse(IntegerToSequence(n,9)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

Flatten@ IntegerDigits[ Range@ 55, 9] (* 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[#, 9] &, 105] (* Robert G. Wilson v, Jul 01 2014 *)

from itertools import count, chain, islice
from sympy.ntheory.factor_ import digits
def A031076_gen(): return chain.from_iterable(digits(m,9)[1:] for m in count(1))
A031076_list = list(islice(A031076_gen(),30)) # Chai Wah Wu, Jan 07 2022

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

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

cp's OEIS Frontend

A030190 Binary Champernowne sequence (or word): write the numbers 0,1,2,3,4,... in base 2 and juxtapose.

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A030302 Write n in base 2 and juxtapose; irregular table in which row n lists the binary expansion of n.

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A030548 Write n in base 6 and juxtapose.

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A031219 Write n in base 5 and juxtapose.

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A003137 Write n in base 3 and juxtapose.

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A030373 Write n in base 4 and juxtapose.

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A031076 Write n in base 9 and juxtapose.