A110625 -id:A110625 - 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

A094640 Decimal expansion of the "alternating Euler constant" log(4/Pi).

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow and Robert G. Wilson v, May 18 2004

Decimal expansion of Sum_{n>=1} (-1)^{n-1} (1/n - log(1 + 1/n)) (see Sondow 2005), so in comparison to A001620's sum formula, log(4/Pi) is an "alternating Euler constant."

			log(4/Pi) = 0.24156447527...

George Boros and Victor Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, Chap. 7.
Jonathan Borwein and Peter Borwein, Pi and the AGM, John Wiley & Sons, New York, 1987, Chap. 11.

G. C. Greubel, Table of n, a(n) for n = 0..10000
M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.
Dirk Huylebrouck, Similarities in irrationality proofs for Pi, ln2, zeta(2) and zeta(3), arXiv:math/0211148 [math.CA], 2002-2004; Amer. Math. Monthly 108 (2001), 222-231.
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.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006; J. Math. Anal. Appl. 332(1) (2007), 292-314.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.
Eric Weisstein's World of Mathematics, Hadjicostas's Formula.
Eric Weisstein's World of Mathematics, Digit Count.

Cf. A094641, A103130, A110625, A110626.

SetDefaultRealField(RealField(100)); R:= RealField(); Log(4/Pi(R)); // G. C. Greubel, Aug 28 2018

Mathematica
```
RealDigits[ Log[4/Pi], 10, 111][[1]]
```

log(4/Pi) \\ Charles R Greathouse IV, Jun 06 2011

Equals Integral_{x=0..1, y=0..1} (x-1)/((1+x*y)*log(x*y)). (see Sondow 2005).
Equals -Integral_{x=0..1} (1-x)^2 dx/((1+x^2)*log(x)). - Amiram Eldar, Jun 29 2020
From Petros Hadjicostas, Jun 29 2020: (Start)
Equals Integral_{x=0..1} (1 - x + log(x))/((1 + x)*log(x)) dx. (Let u = x*y and v = y in Sondow's double integral and integrate w.r.t. v.)
Equals Integral_{x=0..1, y=0..1} (1 - x*y)^2/((1 + x^2*y^2)*(log(x*y))^2). (Apply Glasser's (2019) Theorem 1 on Amiram Eldar's integral above.) (End)
Equals Integral_{0..Pi/2} (sec(t)-2/(Pi-2*t)) dt. - Clark Kimberling, Jul 10 2020
Equals -Sum_{k>=1} log(1 - 1/(2*k+1)^2). - Amiram Eldar, Jul 06 2023

A110626 Denominator of b(n) = -Sum_{k=1..n} A037861(k)/((2k)(2*k+1)), where A037861(k) = (number of 0's) - (number of 1's) in the binary representation of k.

Original entry on oeis.org

6, 6, 14, 504, 27720, 360360, 360360, 765765, 765765, 765765, 1601145, 1601145, 369495, 3061530, 94907430, 16703707680, 116925953760, 4326260289120, 1068586291412640, 43812037947918240, 1883917631760484320

Offset: 1

Jonathan Sondow, Aug 01 2005

Denominators of partial sums of a series for the "alternating Euler constant" log(4/Pi) (see A094640 and Sondow 2005, 2010). Numerators are A110625.

			a(3) = 14 because b(3) = 1/6 + 0 + 1/21 = 3/14.
The first few fractions b(n) are 1/6, 1/6, 3/14, 101/504, 5807/27720, 77801/360360, 82949/360360, ... = A110625/A110626. - _Petros Hadjicostas_, May 15 2020

Petros Hadjicostas, Table of n, a(n) for n = 1..100
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.
Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, 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.
Jonathan Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), 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.

Cf. A037861, A073099, A094640, A110625.

a(n) = denominator(-sum(k=1, n, (#binary(k) - 2*hammingweight(k))/(2*k*(2*k+1))));\\ Petros Hadjicostas, May 15 2020

Lim_{n -> infinity} b(n) = log 4/Pi = 0.24156...

A126388 Denominators in a series for the "alternating Euler constant" log(4/Pi).

Original entry on oeis.org

2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 22, 23, 26, 27, 28, 29, 30, 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, 63, 64, 65, 66, 67, 68, 69, 72, 73, 78, 79, 80, 81, 86, 87, 90, 91, 92

Offset: 2

Jonathan Sondow, Jan 01 2007

All n > 1 such that (# of 1's) != (# of 0's) in the base 2 expansion of floor(n/2). The numerators of the series are A126389.

			floor(13/2) = 6 = 110 base 2, which has (# of 1's) = 2 != 1 = (#
of 0's), so 13 is a member.

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.
Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, 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.
Jonathan Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), 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.
Eric Weisstein's MathWorld, Digit Count.

Complementary to A066879.

Cf. A037861, A094640, A094641, A110625, A110626, A126389.

b[n_] := DigitCount[n,2,1] - DigitCount[n,2,0]; L = {}; Do[If[b[Floor[n/2]] != 0, L = Append[L,n]], {n,2,100}]; L

log(4/Pi) = 1/2 - 1/3 + 2/6 - 2/7 - 1/8 + 1/9 + 1/10 - 1/11 + 1/12 - 1/13 + 3/14 - 3/15 - 2/16 + 2/17 + 2/22 - ...

A126389 Numerators in a series for the "alternating Euler constant" log(4/Pi).

Original entry on oeis.org

1, -1, 2, -2, -1, 1, 1, -1, 1, -1, 3, -3, -2, 2, 2, -2, 2, -2, 2, -2, 4, -4, -3, 3, -1, 1, -1, 1, 1, -1, -1, 1, 1, -1, 1, -1, 3, -3, -1, 1, 1, -1, 1, -1, 3, -3, 1, -1, 3, -3, 3, -3, 5, -5, -4, 4, -2, 2, -2, 2, -2, 2, 2, -2, -2, 2, 2, -2, 2, -2, 2, -2, 4, -4, -2, 2

Offset: 2

Jonathan Sondow, Jan 01 2007

Nonzero values of (-1)^n*b(floor(n/2)) for n > 1, where b(n) = (# of 1's) - (# of 0's) in the base 2 expansion of n. The denominators of the series are A126388.

			floor(15/2) = 7 = 111 base 2, which has (# of 1's) - (# of 0's) = 3, so (-1)^15*3 = -3 is a term.

J. Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, Amer. Math. Monthly 112 (2005) 61-65.
J. Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), 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.
Eric Weisstein's MathWorld, Digit Count

Cf. A037861, A066879, A094640, A094641, A110625, A110626, A126388.

b[n_] := DigitCount[n,2,1] - DigitCount[n,2,0]; L = {}; Do[If[b[Floor[n/2]] != 0, L = Append[L,(-1)^n*b[Floor[n/2]]]], {n,2,100}]; L

Log(4/Pi) = 1/2 - 1/3 + 2/6 - 2/7 - 1/8 + 1/9 + 1/10 - 1/11 + 1/12 - 1/13 + 3/14 - 3/15 - 2/16 + 2/17 + 2/22 - ...

cp's OEIS Frontend

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

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A094640 Decimal expansion of the "alternating Euler constant" log(4/Pi).

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A110626 Denominator of b(n) = -Sum_{k=1..n} A037861(k)/((2*k)*(2*k+1)), where A037861(k) = (number of 0's) - (number of 1's) in the binary representation of k.

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A126388 Denominators in a series for the "alternating Euler constant" log(4/Pi).

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Author

Keywords

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Examples

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Crossrefs

Programs

Mathematica

Formula

A126389 Numerators in a series for the "alternating Euler constant" log(4/Pi).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A110626 Denominator of b(n) = -Sum_{k=1..n} A037861(k)/((2k)(2*k+1)), where A037861(k) = (number of 0's) - (number of 1's) in the binary representation of k.