A076478 -id:A076478 - cp's OEIS frontend

A206446 Positions of 1 in A076478; complement of A206445.

2, 6, 7, 9, 10, 16, 18, 21, 22, 23, 26, 28, 29, 30, 32, 33, 34, 42, 45, 49, 50, 52, 56, 58, 60, 61, 64, 65, 66, 67, 71, 74, 75, 77, 79, 81, 82, 83, 84, 87, 88, 90, 91, 92, 93, 95, 96, 97, 98, 108, 112, 117, 118, 121, 126, 128, 131, 132, 136, 137, 138, 140, 145

Offset: 1

Clark Kimberling, Feb 08 2012

nonn

			The positions of 1 in 0,1,0,0,0,1,1,0,1,1,0,0,0,0,0,1,...
are 2,6,7,9,10,16,...

Cf. A076478, A206445.

d[n_] := Rest@IntegerDigits[n + 1, 2] + 1;
t = -1 + Flatten[Array[d, 100]]
Flatten[Position[t, 0]]  (* A206445 *)
Flatten[Position[t, 1]]  (* A206446 *)

A277914 a(n) is the number of zeros minus the number of ones among the first n terms of A076478.

Original entry on oeis.org

0, 1, 0, 1, 2, 3, 2, 1, 2, 1, 0, 1, 2, 3, 4, 5, 4, 5, 4, 5, 6, 5, 4, 3, 4, 5, 4, 5, 4, 3, 2, 3, 2, 1, 0, 1, 2, 3, 4, 5, 6, 7, 6, 7, 8, 7, 8, 9, 10, 9, 8, 9, 8, 9, 10, 11, 10, 11, 10, 11, 10, 9, 10, 11, 10, 9, 8, 7, 8, 9, 10, 9, 10, 11, 10, 9, 10, 9, 10, 9, 10

Offset: 0

Rémy Sigrist, Nov 03 2016

A036799 gives the positions of zeros in this sequence.

Between two zeros, the graph of the sequence looks like a Takagi (or blancmange) curve.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Wikipedia, Blancmange curve

Cf. A076478, A036799.

Join[{0},Flatten[Array[Accumulate[Flatten[Tuples[{1,-1},#]]]&,5]]] (* Paolo Xausa, Dec 08 2023 *)

a(n) = Sum_{i=0..n-1} (-1)^A076478(i) for any n>=0.

A346306 Position in A076478 of the binary complement of the n-th word in A076478.

Original entry on oeis.org

2, 1, 6, 5, 4, 3, 14, 13, 12, 11, 10, 9, 8, 7, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 126, 125, 124, 123

Offset: 1

Clark Kimberling, Aug 16 2021

Permutation of the positive integers obtained by reversing their order within successive subsets of length 2, 4, 8, 16, ... - Paolo Xausa, Mar 09 2023

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(3) = 6.
From _Paolo Xausa_, Mar 09 2023: (Start)
Written as an irregular triangle, where row r >= 1 has length 2^r and row sum is A103897(r), the sequence begins:
   2,  1;
   6,  5,  4,  3;
  14, 13, 12, 11, 10,  9,  8,  7;
  30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15;
  ... (End)

Michael S. Branicky, Table of n, a(n) for n = 1..16382 (for all words with length <= 13)
Index entries for sequences that are permutations of the natural numbers

Cf. A007931, A076478, A103897, A171757, A346303, A346304.

(See A007931.)
A346306[rowmax_]:=Table[Range[2^(r+1)-2,2^r-1,-1],{r,rowmax}]; A346306[6] (* Paolo Xausa, Mar 09 2023 *)

from itertools import product
def comp(s): z, o = ord('0'), ord('1'); return s.translate({z:o, o:z})
def wgen(maxdigits):
    for digits in range(1, maxdigits+1):
        for b in product("01", repeat=digits):
            yield "".join(b)
def auptod(maxdigits):
    w = [None] + [wn for wn in wgen(maxdigits)]
    return [w.index(comp(w[n])) for n in range(1, 2**(maxdigits+1) - 1)]
print(auptod(6)) # Michael S. Branicky, Sep 03 2021

a(n) = 3*(2^d - 1) - n, where 2^d - 1 <= n <= 2^(d+1) - 2. - Michael S. Branicky, Sep 03 2021

A346309 Positions of words in A076478 such that #0's - #1's is odd.

Original entry on oeis.org

1, 2, 7, 8, 9, 10, 11, 12, 13, 14, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145

Offset: 1

Clark Kimberling, Aug 28 2021

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(3) = 7.

Cf. A007931, A076478, A346310 (complement), A053754.

Mathematica
```
(See A076478)
```

a(n) = A053754(n+1) + 1, conjectured.

A346310 Positions of words in A076478 such that #0's - #1's is even.

Original entry on oeis.org

3, 4, 5, 6, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106

Offset: 1

Clark Kimberling, Aug 28 2021

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(1) = 3.

Cf. A007931, A076478, A346309 (complement), A053738.

Mathematica
```
(See A076478.)
```

a(n) = A053738(n+1) - 1, conjectured.

A271728 Numbers n such that A076478(n)=1.

Original entry on oeis.org

1, 5, 6, 8, 9, 15, 17, 20, 21, 22, 25, 27, 28, 29, 31, 32, 33, 41, 44, 48, 49, 51, 55, 57, 59, 60, 63, 64, 65, 66, 70, 73, 74, 76, 78, 80, 81, 82, 83, 86, 87, 89, 90, 91, 92, 94, 95, 96, 97, 107, 111, 116, 117, 120, 125, 127, 130

Offset: 1

Gabriel Conant, Apr 13 2016

nonn

By definition, this sequence enumerates a 'universal' set S of nonnegative integers: meaning that every finite binary sequence occurs as a consecutive subsequence of the characteristic function of S (viewed as an infinite binary sequence). The infinite graph with vertex set the integers and edge relation '|x-y| in S' is a countable random graph.

Gabriel Conant, Table of n, a(n) for n = 1..3000
P. J. Cameron, The random graph, The Mathematics of Paul Erdos, 2nd ed., Algorithms Combin., 14 (1997), 333-351.

Cf. A076478.

A346307 Number of runs in the n-th word in A076478.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Aug 16 2021

			The first fourteen words w(n) are 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, 100, 101, 110, 111, so that a(9) = 3.

Cf. A007931, A076478.

Mathematica
```
(See A007931.)
```

A362599 The terms of the n-th row of A076478 first appear from position a(n) in A362240.

Original entry on oeis.org

1, 2, 3, 1, 2, 5, 7, 3, 1, 4, 2, 10, 5, 12, 16, 7, 8, 3, 1, 9, 4, 11, 6, 2, 20, 10, 5, 24, 13, 12, 28, 16, 7, 33, 38, 8, 3, 43, 26, 1, 19, 9, 4, 23, 44, 11, 15, 6, 37, 2, 25, 20, 22, 10, 5, 36, 24, 46, 13, 45, 12, 53, 28, 31, 16, 32, 58, 7, 33, 42, 38, 64, 18

Offset: 0

Rémy Sigrist, Apr 27 2023

nonn

			The first terms, alongside the corresponding row of A076478, are:
  n   a(n)  n-th row of A076478
  --  ----  --------------------
   0     1  0
   1     2   1
   2     3    00
   3     1  01
   4     2   10
   5     5      11
   6     7        000
   7     3    001
   8     1  010
   9     4     011
  10     2   100
   .     .  .
  A362240   010011000...

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, Perl program

Cf. A076478, A362240.

Perl
```
See Links section.
```

A308705 Decimal expansion of the constant formed by concatenating the terms of A076478 (binary vectors of lengths 1, 2, 3, ... in numerical order).

Original entry on oeis.org

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

Offset: 0

David McFadzean, Jun 18 2019

The binary vectors "0,1,00,01,10,11,000,001,..." are concatenated into a constant "0.0100011011000001...", then converted to decimal.

			0.276387117279486523734198676211901230555089988160685506143676819115...

Cf. A076478.

m = 100; d[n_] := Rest@IntegerDigits[n + 1, 2] + 1; v = Flatten[Array[d, 4 m]] - 1; RealDigits[FromDigits[v, 2]/2^Length[v], 10, m][[1]] (* Amiram Eldar, Jul 05 2019 after Clark Kimberling at A076478 *)

from bigfloat import *
import string
def GenerateBitstring(bitstring, suffix, recurse):
    if recurse == 0:
        bitstring = bitstring + suffix
    else:
        bitstring = GenerateBitstring(bitstring, suffix + "0", recurse-1)
        bitstring = GenerateBitstring(bitstring, suffix + "1", recurse-1)
    return bitstring
VulcanBinary = ""
MaxRecursion = 8
for i in range(1,MaxRecursion+1):
    VulcanBinary = GenerateBitstring(VulcanBinary, "", i)
print('.' + VulcanBinary)
with precision(2000):
    VulcanDecimal = BigFloat(0)
    b = BigFloat(1)
    for c in VulcanBinary:
        b = b/2.
        if c == '1':
            VulcanDecimal = VulcanDecimal+b
print(VulcanDecimal)
print(string.join(x + ',' for x in str(VulcanDecimal)[2:]))

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

Original entry on oeis.org

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

cp's OEIS Frontend

A206446 Positions of 1 in A076478; complement of A206445.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A277914 a(n) is the number of zeros minus the number of ones among the first n terms of A076478.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A346306 Position in A076478 of the binary complement of the n-th word in A076478.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A346309 Positions of words in A076478 such that #0's - #1's is odd.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A346310 Positions of words in A076478 such that #0's - #1's is even.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A271728 Numbers n such that A076478(n)=1.

Views

Author

Keywords

Comments

Links

Crossrefs

A346307 Number of runs in the n-th word in A076478.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A362599 The terms of the n-th row of A076478 first appear from position a(n) in A362240.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Perl

A308705 Decimal expansion of the constant formed by concatenating the terms of A076478 (binary vectors of lengths 1, 2, 3, ... in numerical order).

Views

Author

Keywords

Comments

Examples

Crossrefs