A110625 Numerator 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.

1, 1, 3, 101, 5807, 77801, 82949, 170636, 170636, 170636, 363113, 363113, 84848, 710567, 22435781, 3901243741, 27210449083, 1003538672911, 248595095590537, 10165684261926701, 438167567023512863, 439119040574907047

Offset

1,3

Comments

Numerators of partial sums of a series for the "alternating Euler constant" log(4/Pi) (see A094640 and Sondow 2005, 2010). Denominators are A110626.

Links

Petros Hadjicostas, Table of n, a(n) for n = 1..120

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) = 3 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) = numerator(-sum(k=1, n, (#binary(k) - 2*hammingweight(k))/(2*k*(2*k+1)))); \\ Petros Hadjicostas, May 15 2020

Crossrefs

Cf. A037861, A073099, A094640, A110626.

Sequence in context: A336437 A069457 A142416 * A108220 A130733 A037062

Adjacent sequences: A110622 A110623 A110624 * A110626 A110627 A110628

Keyword

easy,frac,nonn

Author

Jonathan Sondow, Aug 01 2005

Status

approved