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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. 6
1, 1, 3, 101, 5807, 77801, 82949, 170636, 170636, 170636, 363113, 363113, 84848, 710567, 22435781, 3901243741, 27210449083, 1003538672911, 248595095590537, 10165684261926701, 438167567023512863, 439119040574907047 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 14 05:20 EDT 2022. Contains 356110 sequences. (Running on oeis4.)