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

%I A030190 #93 Feb 16 2025 08:32:35
%S A030190 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,
%T A030190 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,
%U A030190 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
%N A030190 Binary Champernowne sequence (or word): write the numbers 0,1,2,3,4,... in base 2 and juxtapose.
%C A030190 a(A003607(n)) = 0 and for n > 0: a(A030303(n)) = 1. - _Reinhard Zumkeller_, Dec 11 2011
%C A030190 An irregular table in which the n-th row lists the bits of n (see the example section). - _Jason Kimberley_, Dec 07 2012
%C A030190 The binary Champernowne constant: it is normal in base 2. - _Jason Kimberley_, Dec 07 2012
%C A030190 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
%D A030190 Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
%H A030190 Reinhard Zumkeller, <a href="/A030190/b030190.txt">Table of n, a(n) for n = 0..10000</a>
%H A030190 Jean Berstel, <a href="http://www-igm.univ-mlv.fr/~berstel/">Home Page</a> (in case the following link should be broken)
%H A030190 Jean Berstel and Juhani Karhumäki, <a href="http://www-igm.univ-mlv.fr/~berstel/Articles/2003TutorialCoWdec03.pdf">Combinatorics on words-a tutorial</a>. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, # 79, pp. 178-228, 2003.
%H A030190 S. Ferenczi, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00400-2">Complexity of sequences and dynamical systems</a>, Discrete Math., 206 (1999), 145-154.
%H A030190 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BinaryChampernowneConstant.html">Binary Champernowne Constant</a>
%H A030190 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%e A030190 As an array, this begins:
%e A030190 0,
%e A030190 1,
%e A030190 1, 0,
%e A030190 1, 1,
%e A030190 1, 0, 0,
%e A030190 1, 0, 1,
%e A030190 1, 1, 0,
%e A030190 1, 1, 1,
%e A030190 1, 0, 0, 0,
%e A030190 1, 0, 0, 1,
%e A030190 1, 0, 1, 0,
%e A030190 1, 0, 1, 1,
%e A030190 1, 1, 0, 0,
%e A030190 1, 1, 0, 1,
%e A030190 1, 1, 1, 0,
%e A030190 1, 1, 1, 1,
%e A030190 1, 0, 0, 0, 0,
%e A030190 1, 0, 0, 0, 1,
%e A030190 ...
%t A030190 Flatten[ Table[ IntegerDigits[n, 2], {n, 0, 26}]] (* _Robert G. Wilson v_, Mar 08 2005 *)
%t A030190 First[RealDigits[ChampernowneNumber[2], 2, 100, 0]] (* _Paolo Xausa_, Jun 16 2024 *)
%o A030190 (Haskell)
%o A030190 import Data.List (unfoldr)
%o A030190 a030190 n = a030190_list !! n
%o A030190 a030190_list = concatMap reverse a030308_tabf
%o A030190 -- _Reinhard Zumkeller_, Jun 16 2012, Dec 11 2011
%o A030190 (Magma) [0]cat &cat[Reverse(IntegerToSequence(n,2)):n in[1..31]]; // _Jason Kimberley_, Dec 07 2012
%o A030190 (PARI) A030190_row(n)=if(n,binary(n),[0]) \\ _M. F. Hasler_, Oct 12 2020
%o A030190 (Python)
%o A030190 from itertools import count, islice
%o A030190 def A030190_gen(): return (int(d) for m in count(0) for d in bin(m)[2:])
%o A030190 A030190_list = list(islice(A030190_gen(),30)) # _Chai Wah Wu_, Jan 07 2022
%Y A030190 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_.
%Y A030190 a(n) = T(A030530(n), A083652(A030530(n))-n-1), T as defined in A083651, a(A083652(k))=1.
%Y A030190 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
%Y A030190 A076478 is a similar sequence.
%Y A030190 For run lengths see A056062; see also A318924.
%Y A030190 See also A066099 for (run lengths of 0s) + 1 = first difference of positions of 1s given by A030303.
%K A030190 nonn,base,cons,easy,tabf
%O A030190 0,1
%A A030190 _Clark Kimberling_