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A030190 Binary Champernowne sequence (or word): write the numbers 0,1,2,3,4,... in base 2 and juxtapose. 89
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
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
REFERENCES
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
LINKS
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
EXAMPLE
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,
...
MATHEMATICA
Flatten[ Table[ IntegerDigits[n, 2], {n, 0, 26}]] (* Robert G. Wilson v, Mar 08 2005 *)
PROG
(Haskell)
import Data.List (unfoldr)
a030190 n = a030190_list !! n
a030190_list = concatMap reverse a030308_tabf
-- Reinhard Zumkeller, Jun 16 2012, Dec 11 2011
(Magma) [0]cat &cat[Reverse(IntegerToSequence(n, 2)):n in[1..31]]; // Jason Kimberley, Dec 07 2012
(PARI) A030190_row(n)=if(n, binary(n), [0]) \\ M. F. Hasler, Oct 12 2020
(Python)
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
CROSSREFS
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.
Sequence in context: A014578 A323153 A288861 * A353471 A157658 A296211
KEYWORD
nonn,base,cons,easy,tabf
AUTHOR
STATUS
approved

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Last modified March 28 05:01 EDT 2024. Contains 371235 sequences. (Running on oeis4.)