A072603 -id:A072603 - 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

A072601 Numbers which in base 2 have at least as many 1's as 0's.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 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

Offset: 1

Reinhard Zumkeller, Jun 23 2002

			8 = 1000_2 is not present (one '1', three '0's).
10 is present because 10=1010_2 contains 2 '0's and 2 '1's: 2<=2;
11 is present because 11=1011_2 contains 1 '0' and 3 '1's: 1<=3.

T. D. Noe, Table of n, a(n) for n = 1..5370 (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, A061854.
Cf. A037861(a(n)) <= 0.
Cf. A072600 (#0's < #1's), this seq (#0's <= #1's), A031443 (#0's = #1's).
Cf. A072602 (#0's >= #1's), A072603 (#0's > #1's), A044951 (#0's <> #1's).

a072601 n = a072601_list !! (n-1)
a072601_list = filter ((<= 0) . a037861) [0..]
-- Reinhard Zumkeller, Aug 01 2013

geQ[n_] := Module[{a, b}, {a, b} = DigitCount[n, 2]; a >= b]; Select[Range[103], geQ] (* T. D. Noe, Apr 20 2013 *)
Select[Range[110],DigitCount[#,2,1]>=DigitCount[#,2,0]&] (* Harvey P. Dale, Aug 12 2023 *)

is(n)=2*hammingweight(n)>exponent(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 *)

A352546 Numbers having more even than odd digits when written in base 10.

Original entry on oeis.org

0, 2, 4, 6, 8, 20, 22, 24, 26, 28, 40, 42, 44, 46, 48, 60, 62, 64, 66, 68, 80, 82, 84, 86, 88, 100, 102, 104, 106, 108, 120, 122, 124, 126, 128, 140, 142, 144, 146, 148, 160, 162, 164, 166, 168, 180, 182, 184, 186, 188, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 212

Offset: 1

M. F. Hasler, Jul 03 2022

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A196563, A196564.
Cf. A072603 (same in base 2).
Cf. A117076 (subsequence of primes).
Cf. A352547 (numbers having more odd than even decimal digits).

A352546Q[k_] := Length[#] > 2*Count[#, _?OddQ] & [IntegerDigits[k]];
Select[Range[0, 300], A352546Q] (* Paolo Xausa, Nov 28 2024 *)

select( {is_A352546(n)=vecsum(n=digits(n)%2)*2<#n+!n}, [0..222])

def ok(n): return len(s:=str(n)) < 2*sum(1 for c in s if c in "02468")
print([k for k in range(213) if ok(k)]) # Michael S. Branicky, Jul 03 2022

A355274 Numbers having more even than odd digits when written in base 3.

Original entry on oeis.org

0, 2, 6, 8, 9, 11, 15, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 29, 33, 35, 45, 47, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 65, 69, 71, 72, 73, 74, 75, 77, 78, 79, 80, 81, 82, 83, 84, 86, 87, 88, 89, 90, 92, 96, 98, 99, 100, 101, 102, 104, 105, 106, 107, 108, 110, 114, 116

Offset: 1

M. F. Hasler, Jul 03 2022

Cf. A072603 (same in base 2), A352546 (same in base 10).

select( {is_A355274(n)=vecsum(n=digits(n,3)%2)*2<#n+!n}, [0..123])

from sympy.ntheory import digits
def ok(n):
    d = digits(n, 3)[1:]
    return len(d) < 2*sum(1 for di in d if di%2 == 0)
print([k for k in range(117) if ok(k)]) # Michael S. Branicky, Jul 03 2022

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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A072601 Numbers which in base 2 have at least as many 1's as 0'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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A352546 Numbers having more even than odd digits when written in base 10.

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