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

1,1

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.

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.

Formula

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

Example

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

Prog

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

Crossrefs

Cf. A037861, A073099, A094640, A110625.

Sequence in context: A315806 A315807 A315808 * A339721 A341832 A072695

Adjacent sequences: A110623 A110624 A110625 * A110627 A110628 A110629

Keyword

easy,frac,nonn

Author

Jonathan Sondow, Aug 01 2005

Status

approved