A030530 -id:A030530 - 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

A083652 Sum of lengths of binary expansions of 0 through n.

Original entry on oeis.org

1, 2, 4, 6, 9, 12, 15, 18, 22, 26, 30, 34, 38, 42, 46, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 136, 142, 148, 154, 160, 166, 172, 178, 184, 190, 196, 202, 208, 214, 220, 226, 232, 238, 244, 250, 256, 262, 268, 274, 280, 286, 292

Offset: 0

Reinhard Zumkeller, May 01 2003

a(n) = A001855(n) + 1 for n > 0;
a(0) = A070939(0)=1, n > 0: a(n) = a(n-1) + A070939(n).
A030190(a(k))=1; A030530(a(k)) = k + 1.
Partial sums of A070939. - Hieronymus Fischer, Jun 12 2012
Young writes "If n = 2^i + k [...] the maximum is (i+1)(2^i+k)-2^{i+1}+2." - Michael Somos, Sep 25 2012

			G.f. = 1 + 2*x + 4*x^2 + 6*x^3 + 9*x^4 + 12*x^5 + 15*x^6 + 18*x^7 + 22*x^8 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and asymptotic solutions of the recurrence f(n) = f(floor(n/2)) + f(ceiling(n/2)) + g(n): theory and applications, Preprint 2016.
Hsien-Kuei Hwang, S. Janson, and T.-H. Tsai, Exact and Asymptotic Solutions of a Divide-and-Conquer Recurrence Dividing at Half: Theory and Applications, ACM Transactions on Algorithms, 13:4 (2017), #47; DOI: 10.1145/3127585.
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 36.
Alfred Young, The Maximum Order of an Irreducible Covariant of a System of Binary Forms, Proc. Roy. Soc. 72 (1903), 399-400 = The Collected Papers of Alfred Young, 1977, 136-137.
Index entries for sequences related to binary expansion of n

Cf. A000120, A007088, A023416, A059015, A070939 (base 2), A123753.

A296349 is an essentially identical sequence.

a083652 n = a083652_list !! n
a083652_list = scanl1 (+) a070939_list
-- Reinhard Zumkeller, Jul 05 2012

Accumulate[Length/@(IntegerDigits[#,2]&/@Range[0,60])] (* Harvey P. Dale, May 28 2013 *)
a[n_] := (n + 1) IntegerLength[n + 1, 2] - 2^IntegerLength[n + 1, 2] + 2;Table[a[n], {n, 0, 58}] (* Peter Luschny, Dec 02 2017 *)

{a(n) = my(i); if( n<0, 0, n++; i = length(binary(n)); n*i - 2^i + 2)}; /* Michael Somos, Sep 25 2012 */

a(n)=my(i=#binary(n++));n*i-2^i+2 \\ equivalent to the above

def A083652(n):
    s, i, z = 1, n, 1
    while 0 <= i: s += i; i -= z; z += z
    return s
print([A083652(n) for n in range(0, 59)]) # Peter Luschny, Nov 30 2017

def A083652(n): return 2+(n+1)*(m:=(n+1).bit_length())-(1<Chai Wah Wu, Mar 01 2023

a(n) = 2 + (n+1)*ceiling(log_2(n+1)) - 2^ceiling(log_2(n+1)).
G.f.: g(x) = 1/(1-x) + (1/(1-x)^2)*Sum_{j>=0} x^2^j. - Hieronymus Fischer, Jun 12 2012; corrected by Ilya Gutkovskiy, Jan 08 2017
a(n) = A123753(n) - n. - Peter Luschny, Nov 30 2017

A100922 k appears A000120(k) times (appearances equal number of 1-bits).

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 7, 8, 9, 9, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 17, 17, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 27, 28, 28, 28, 29, 29, 29, 29, 30, 30, 30, 30

Offset: 0

Rick L. Shepherd, Nov 21 2004

Clearly every positive integer appears at least once in this sequence.

			The binary representation of 16 is 10000, which has one 1-bit (and four 0-bits), hence 16 appears once in this sequence (and four times in A100921).

Antti Karttunen, Table of n, a(n) for n = 0..11264

Cf. A100921 (n's appearances equal its number of 0-bits), A030530 (n's appearances equal its total number of bits), A227737 (n's appearances equal its total number of runs), A000069, A000120, A000788, A163510, A243067.

T:= n-> n$add(i, i=Bits[Split](n)):
seq(T(n), n=1..30);  # Alois P. Heinz, Nov 11 2024

Table[Table[n,DigitCount[n,2,1]],{n,30}]//Flatten (* Harvey P. Dale, Aug 31 2017 *)

def A000788(n): return (n+1)*n.bit_count()+(sum((m:=1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1))>>1)
def A100922(n):
    if n == 0: return 1
    m, k = 1, 1
    while A000788(m)<=n: m<<=1
    while m-k>1:
        r = m+k>>1
        if A000788(r)>n:
            m = r
        else:
            k = r
    return m # Chai Wah Wu, Nov 11 2024

a(n) = the least k such that A000788(k) > n. - Antti Karttunen, Jun 20 2014

Sum_{n>=1} (-1)^(n+1)/a(n) = Sum_{n>=1} (-1)^(n+1)/A000069(n) = 0.67968268... . - Amiram Eldar, Feb 18 2024

A100921 n appears A023416(n) times (appearances equal number of 0-bits).

Original entry on oeis.org

0, 2, 4, 4, 5, 6, 8, 8, 8, 9, 9, 10, 10, 11, 12, 12, 13, 14, 16, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 23, 24, 24, 24, 25, 25, 26, 26, 27, 28, 28, 29, 30, 32, 32, 32, 32, 32, 33, 33, 33, 33, 34, 34, 34, 34, 35, 35, 35, 36, 36, 36, 36, 37, 37, 37

Offset: 0

Rick L. Shepherd, Nov 21 2004

			The binary representation of 16 is 10000, which has four 0-bits (and one 1-bit), hence 16 appears four times in this sequence (but only once in A100922).

Cf. A100922 (n's appearances equal its number of 1-bits), A030530 (n's appearances equal its total number of bits), A023416, A059009.

Flatten[Table[Table[n, {DigitCount[n, 2, 0]}], {n, 0, 37}]] (* Amiram Eldar, Feb 18 2024 *)

def A059015(n): return 2+(n+1)*((t:=(n+1).bit_length())-n.bit_count())-(1<>j)-(r if n<<1>=m*(r:=k<<1|1) else 0)) for j in range(1,n.bit_length()+1))>>1)
def A100921(n):
    if n == 0: return 0
    m, k = 1, 1
    while A059015(m)<=n: m<<=1
    while m-k>1:
        r = m+k>>1
        if A059015(r)>n:
            m = r
        else:
            k = r
    return m  # Chai Wah Wu, Nov 11 2024

Sum_{n>=1} (-1)^(n+1)/a(n) = Sum_{n>=1} (-1)^(n+1)/A059009(n) = 0.395592509... . - Amiram Eldar, Feb 18 2024

A100470 n appears A055642(n) times (appearances equal number of decimal digits).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38, 39, 39, 40, 40, 41

Offset: 0

Rick L. Shepherd, Nov 21 2004

Lexicographically smallest nondecreasing left inverse to A117804: a(A117804(n)) = n. Gives the number to which belongs the digit A007376(n). - M. F. Hasler, Oct 23 2019

Cf. A055642, A030530 (n's appearances equal its number of bits).

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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A083652 Sum of lengths of binary expansions of 0 through n.

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A100922 k appears A000120(k) times (appearances equal number of 1-bits).

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A100921 n appears A023416(n) times (appearances equal number of 0-bits).

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A100470 n appears A055642(n) times (appearances equal number of decimal digits).

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