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.

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

Offset: 1

Author

Jonathan Sondow, Aug 01 2005

Keywords

• easy
• frac
• nonn

Comments

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

Examples

			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
		

Links

• 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.

Crossrefs

Cf. A037861, A073099, A094640, A110625.

Programs

PARI

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

Formula

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