A072601 -id:A072601 - cp's OEIS frontend

A037861 (Number of 0's) - (number of 1's) in the base-2 representation of n.

1, -1, 0, -2, 1, -1, -1, -3, 2, 0, 0, -2, 0, -2, -2, -4, 3, 1, 1, -1, 1, -1, -1, -3, 1, -1, -1, -3, -1, -3, -3, -5, 4, 2, 2, 0, 2, 0, 0, -2, 2, 0, 0, -2, 0, -2, -2, -4, 2, 0, 0, -2, 0, -2, -2, -4, 0, -2, -2, -4, -2, -4, -4, -6, 5, 3, 3, 1, 3, 1, 1, -1, 3

Offset: 0

Clark Kimberling

-Sum_{n>=1} a(n)/((2*n)*(2*n+1)) = the "alternating Euler constant" log(4/Pi) = 0.24156... - (see A094640 and Sondow 2005, 2010).

a(A072600(n)) < 0; a(A072601(n)) <= 0; a(A031443(n)) = 0; a(A072602(n)) >= 0; a(A072603(n)) > 0; a(A031444(n)) = 1; a(A031448(n)) = -1; abs(a(A089648(n))) <= 1. - Reinhard Zumkeller, Feb 07 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, arXiv:math/0211148 [math.CA], 2002-2004; Amer. Math. Monthly 112 (2005), 61-65.
Jonathan Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), arXiv:math/0508042 [math.NT], 2005; Additive Number Theory, Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson (D. Chudnovsky and G. Chudnovsky, eds.), Springer, 2010, pp. 331-340.
Ralf Stephan, Divide-and-conquer generating functions. I. Elementary sequences, arXiv:math/0307027 [math.CO], 2003.
Ralf Stephan, Some divide-and-conquer sequences with (relatively) simple ordinary generating functions, 2004.
Ralf Stephan, Table of generating functions (ps file).
Ralf Stephan, Table of generating functions (pdf file).
Index entries for sequences related to binary expansion of n

Cf. A031443 for n when a(n)=0, A053738 for n when a(n) odd, A053754 for n when a(n) even, A030300 for a(n+1) mod 2.
Cf. A066879, A094640, A110625, A145037.
Cf. A072600, A072601, A072602, A072603, A031444, A031448, A089648.
See A268289 for a recurrence based on this sequence.

a037861 n = a023416 n - a000120 n  -- Reinhard Zumkeller, Aug 01 2013

A037861:= proc(n) local L;
     L:= convert(n,base,2);
     numboccur(0,L) - numboccur(1,L)
end proc:
map(A037861, [$0..100]); # Robert Israel, Mar 08 2016

Table[Count[ IntegerDigits[n, 2], 0] - Count[IntegerDigits[n, 2], 1], {n, 0, 75}]

a(n) = if (n==0, 1, 1 + logint(n, 2) - 2*hammingweight(n)); \\ Michel Marcus, May 15 2020 and Jun 16 2020

def A037861(n):
    return 2*format(n,'b').count('0')-len(format(n,'b')) # Chai Wah Wu, Mar 07 2016

From Henry Bottomley, Oct 27 2000: (Start)
a(n) = A023416(n) - A000120(n) = A029837(n) - 2*A000120(n) = 2*A023416(n) - A029837(n).
a(2*n) = a(n) + 1; a(2*n + 1) = a(2*n) - 2 = a(n) - 1. (End)
G.f. satisfies A(x) = (1 + x)*A(x^2) - x*(2 + x)/(1 + x). - Franklin T. Adams-Watters, Dec 26 2006
a(n) = b(n) for n > 0 with b(0) = 0 and b(n) = b(floor(n/2)) + (-1)^(n mod 2). - Reinhard Zumkeller, Dec 31 2007
G.f.: 1 + (1/(1 - x))*Sum_{k>=0} x^(2^k)*(x^(2^k) - 1)/(1 + x^(2^k)). - Ilya Gutkovskiy, Apr 07 2018

A072600 Numbers which in base 2 have fewer 0's than 1's.

Original entry on oeis.org

1, 3, 5, 6, 7, 11, 13, 14, 15, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 39, 43, 45, 46, 47, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 71, 75, 77, 78, 79, 83, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 99, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115

Offset: 1

Reinhard Zumkeller, Jun 23 2002

A037861(a(n)) < 0.

b_k = {a(n) | for all n s.t. a(n) contains k binary digits equal to 1} is the list of all valid win/loss round sequences in a "best of 2k-1" two player game, where 1 is a win and 0 is a loss. For example 19 = 10011b represents a game where the winner won the first two rounds, lost the next two, and won the last one. |b_k| = A001700(k). - Philippe Beaudoin, May 14 2014

			11 is present because '1011' contains 1 '0' and 3 '1's: 1<3.

T. D. Noe, Table of n, a(n) for n = 1..4733 ( numbers < 2^13)
Jason Bell, Thomas Finn Lidbetter, Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arXiv:1804.07996 [cs.FL], 2018.
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
Index entries for sequences related to binary expansion of n

Cf. A007088, A000120, A023416, A072601, A031443, A072602, A072603, A037861

a072600 n = a072600_list !! (n-1)
a072600_list = filter ((< 0) . a037861) [0..]
-- Reinhard Zumkeller, Mar 31 2015

Select[Range[130],DigitCount[#,2,0]Harvey P. Dale, Jan 12 2011 *)

is(n)=2*hammingweight(n)>exponent(n)+1 \\ Charles R Greathouse IV, Apr 18 2020

A061854 Nondiving binary sequences: numbers which in base 2 have at least the same number of 1's as 0's and reading the binary expansion from left (msb) to right (least significant bit), the number of 0's never exceeds the number of 1's.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115, 116

Offset: 1

Antti Karttunen, May 11 2001

"msb" = "most significant bit", A053644.
These encode lattice walks using steps (+1,+1) (= 1's in binary expansion) and (+1,-1) (= 0's in binary expansion) that start from the origin (0,0) and never "dive" under the "sea-level" y=0.
The number of such walks of length n (here: the terms of binary width n) is given by C(n,floor(n/2)) = A001405, which is based on the fact mentioned in Guy's article that the shallow diagonals of the Catalan triangle A009766 sum to A001405.
From Jason Kimberley, Feb 08 2013: (Start)
This sequence is a subsequence of A072601.
Define a map from this set onto the nonnegative integers as follows: set the output bit string to be empty, representing zero; process the input string from left to right; when 1 occurs, change the rightmost 0 in the output to 1; if there is no 0 in the output, prepend a 1; when 0 occurs in the input, change the rightmost 1 in the output to 1. The definition of this sequence ensures that we always have a 1 in the output when a 0 occurs in the input. We this map is onto by showing the restriction to the subset Asubsequence is onto. (End)
The binary representation of a(n) is the numeric representation of the left half of a symmetric balanced string of parentheses with "(" representing 1 and ")" representing 0 (see comments and examples in A001405). Some of the numbers in this sequence cannot be realized as the 1-0-pattern of the odd/even positions of 1's in any row n of A237048 that determines the parts and their widths in the symmetric representation of sigma(n), see A352696. - Hartmut F. W. Hoft, Mar 29 2022

			From _Hartmut F. W. Hoft_, Mar 29 2022: (Start)
The columns in the table are the numbers n, the base-2 representation of n, the left half of the symmetric balanced string of parentheses corresponding to n, validity of the nondiving property for n, and associated number a(n):
1   1      (      True    a(1)
2   10     ()     True    a(2)
3   11     ((     True    a(3)
4   100    ())    False    -
5   101    ()(    True    a(4)
6   110    (()    True    a(5)
7   111    (((    True    a(6)
8   1000   ()))   False    -
9   1001   ())(   False    -
10  1010   ()()   True    a(7)
...
20  10100  ()())  False    -
21  10101  ()()(  True    a(13)
...
(End)

Harvey P. Dale, Table of n, a(n) for n = 1..1000
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Antti Karttunen, Some notes on Catalan's Triangle
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
Index entries for sequences related to binary expansion of n

Cf. A001405, A031443, A036990, A036991, A061855, A014486.

Cf. A237593, A237048, A237591, A237593, A352696.

# We use a simple backtracking algorithm: map(op,[seq(NonDivingLatticeSequences(j),j=1..10)]);
NDLS_GLOBAL := []; NonDivingLatticeSequences := proc(n) global NDLS_GLOBAL; NDLS_GLOBAL := []; NonDivingLatticeSequencesAux(0,0,n); RETURN(NDLS_GLOBAL); end;
NonDivingLatticeSequencesAux := proc(x,h,i) global NDLS_GLOBAL; if(0 = i) then NDLS_GLOBAL := [op(NDLS_GLOBAL),x]; else if(h > 0) then NonDivingLatticeSequencesAux((2*x),h-1,i-1); fi; NonDivingLatticeSequencesAux((2*x)+1,h+1,i-1); fi; end;

a061854[n_] := Select[Range[n], !MemberQ[FoldList[#1+If[#2>0, 1, -1]&, 0, IntegerDigits[n, 2]], -1]]
a061854[116] (* Hartmut F. W. Hoft, Mar 29 2022 *)
Select[Range[120],Min[Accumulate[IntegerDigits[#,2]/.(0->-1)]]>=0&] (* Harvey P. Dale, Sep 11 2023 *)

A072603 Numbers which in base 2 have more 0's than 1's.

Original entry on oeis.org

4, 8, 16, 17, 18, 20, 24, 32, 33, 34, 36, 40, 48, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 76, 80, 81, 82, 84, 88, 96, 97, 98, 100, 104, 112, 128, 129, 130, 131, 132, 133, 134, 136, 137, 138, 140, 144, 145, 146, 148, 152, 160, 161, 162, 164, 168, 176, 192, 193

Offset: 1

Reinhard Zumkeller, Jun 23 2002

			8 is present because '1000' contains 3 '0's and 1 '1': 3>1.

T. D. Noe, Table of n, a(n) for n = 1..5202 (numbers up to 2^14)
Jason Bell, Thomas Finn Lidbetter, Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arXiv:1804.07996 [cs.FL], 2018.
Index entries for sequences related to binary expansion of n

Cf. A007088, A000120, A023416, A072600, A072601, A031443, A072602.

Cf. A037861(a(n)) > 0.

a072603 n = a072603_list !! (n-1)
a072603_list = filter ((> 0) . a037861) [1..]
-- Reinhard Zumkeller, Mar 31 2015

gtQ[n_] := Module[{a, b}, {a, b} = DigitCount[n, 2]; b > a]; Select[Range[2^8], gtQ] (* T. D. Noe, Apr 20 2013 *)
Select[Range[200],DigitCount[#,2,0]>DigitCount[#,2,1]&] (* Harvey P. Dale, Feb 26 2023 *)

is(n)=2*hammingweight(n)Charles R Greathouse IV, Apr 18 2020

A072602 Numbers such that in base 2 the number of 0's is >= the number of 1's.

Original entry on oeis.org

2, 4, 8, 9, 10, 12, 16, 17, 18, 20, 24, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 44, 48, 49, 50, 52, 56, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 76, 80, 81, 82, 84, 88, 96, 97, 98, 100, 104, 112, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142

Offset: 1

Reinhard Zumkeller, Jun 23 2002

			8 is present because '1000' contains 3 '0's and 1 '1': 3 >= 1;
9 is present because '1001' contains 2 '0's and 2 '1's: 2 >= 2.

T. D. Noe, Table of n, a(n) for n = 1..7555 (numbers up to 2^14)
Jason Bell, Thomas Finn Lidbetter, Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arXiv:1804.07996 [cs.FL], 2018.
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
Index entries for sequences related to binary expansion of n

Cf. A007088, A000120, A023416, A072600, A072601, A031443, A072603.

Cf. A037861(a(n)) >= 0.

a072602 n = a072602_list !! (n-1)
a072602_list = filter ((>= 0) . a037861) [1..]
-- Reinhard Zumkeller, Mar 31 2015

Select[Range[150],DigitCount[#,2,0]>=DigitCount[#,2,1]&] (* Harvey P. Dale, May 09 2012 *)

is(n)=2*hammingweight(n)<=exponent(n)+1 \\ Charles R Greathouse IV, Apr 18 2020

Edited by N. J. A. Sloane, Jun 23 2009

A044951 Numbers having a different number of ones and zeros in base 2.

Original entry on oeis.org

1, 3, 4, 5, 6, 7, 8, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 39, 40, 43, 45, 46, 47, 48, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Clark Kimberling

			From _Michael De Vlieger_, Feb 07 2019: (Start)
11 (binary 1011) has more 1's than 0's, thus it is in the sequence.
12 (binary 1100) has an equal number of 0's and 1's, thus it is not in the sequence.
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..14031 (all terms k <= 2^14).
Jason Bell, Thomas Finn Lidbetter, and Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arXiv:1804.07996 [cs.FL], 2018.
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
Index entries for sequences related to binary expansion of n

Cf. A072600 (#0's < #1's), A072601 (#0's <= #1's), A031443 (#0's = #1's).

Cf. A072602 (#0's >= #1's), A072603 (#0's > #1's), this sequence (#0's <> #1's).

Select[Range@ 77, UnsameQ @@ DigitCount[#, 2] &] (* Michael De Vlieger, Feb 07 2019 *)

is(n)=2*hammingweight(n)!=exponent(n)+1 \\ Charles R Greathouse IV, Apr 18 2020

a(n) ~ n. - Charles R Greathouse IV, Apr 18 2020

A362030 Irregular triangle read by rows where row n contains the balanced binary words of length 2n interpreted as binary numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 10, 12, 7, 11, 13, 14, 19, 21, 22, 25, 26, 28, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 15, 23, 27, 29, 30, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 71, 75, 77, 78, 83, 85, 86, 89, 90, 92, 99, 101, 102, 105, 106, 108, 113, 114, 116, 120, 135

Offset: 1

Louis Conover, Apr 05 2023

Within a row, strings are ordered lexicographically, which means the resulting values are ordered numerically.
This is from an idea of David Lovler, which he calls "zigzags". It is a rearrangement of A072601. A072603 lists all the numbers that are not in this sequence. A000984 gives the number of coin flip sequences of length 2,4,6, etc.
Not a permutation of the integers. E.g. 8 never occurs. When there are more 0's than 1's, adding 0's doesn't bring it to balance. - Kevin Ryde, Aug 31 2023

			The first few terms written as binary words with leading 0's: 01, 10, 0011, 0101, 0110, 1001, 1010, 1100, 000111, 001011, 001101, 001110, ... (cf. A368804).
Triangle T(n,k) begins:
   1,  2;
   3,  5,  6,  9, 10, 12;
   7, 11, 13, 14, 19, 21, 22, 25, 26, 28, 35, 37, 38, ...;
  15, 23, 27, 29, 30, 39, 43, 45, 46, 51, 53, 54, 57, ...;
  ...

Paolo Xausa, Table of n, a(n) for n = 1..17576 (rows 1..8 of the triangle, flattened).

Columns k=1-2 give: A000225, A083329.
Row sums give A131568.
Main diagonal gives A036563(n+1).
Cf. A000984 (row lengths), A072601, A072603, A368804 (binary).

T:= n-> sort(map(Bits[Join], combinat[permute]([0$n, 1$n])))[]:
seq(T(n), n=1..4);  # Alois P. Heinz, Apr 13 2023

T[n_] := Sort[FromDigits[#, 2] & /@ Permutations[Join[ConstantArray[0, n], ConstantArray[1, n]]]]; Flatten[Table[T[n], {n, 1, 4}]][[1 ;; 64]] (* Robert P. P. McKone, Aug 29 2023 *)

cp's OEIS Frontend

A037861 (Number of 0's) - (number of 1's) in the base-2 representation of n.

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A072600 Numbers which in base 2 have fewer 0's than 1's.

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A061854 Nondiving binary sequences: numbers which in base 2 have at least the same number of 1's as 0's and reading the binary expansion from left (msb) to right (least significant bit), the number of 0's never exceeds the number of 1's.

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A072603 Numbers which in base 2 have more 0's than 1's.

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A072602 Numbers such that in base 2 the number of 0's is >= the number of 1's.

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A044951 Numbers having a different number of ones and zeros in base 2.

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A362030 Irregular triangle read by rows where row n contains the balanced binary words of length 2n interpreted as binary numbers.

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