|
%I #27 May 15 2020 13:05:15
%S 1,1,3,101,5807,77801,82949,170636,170636,170636,363113,363113,84848,
%T 710567,22435781,3901243741,27210449083,1003538672911,248595095590537,
%U 10165684261926701,438167567023512863,439119040574907047
%N 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.
%C Numerators of partial sums of a series for the "alternating Euler constant" log(4/Pi) (see A094640 and Sondow 2005, 2010). Denominators are A110626.
%H Petros Hadjicostas, <a href="/A110625/b110625.txt">Table of n, a(n) for n = 1..120</a>
%H Jonathan Sondow, <a href="https://arxiv.org/abs/math/0211148"> Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula</a>, arXiv:math/0211148 [math.CA], 2002-2004.
%H Jonathan Sondow, <a href="https://www.jstor.org/stable/30037385"> Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula</a>, Amer. Math. Monthly 112 (2005), 61-65.
%H Jonathan Sondow, <a href="https://arxiv.org/abs/math/0508042">New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi)</a>, arXiv:math/0508042 [math.NT], 2005.
%H Jonathan Sondow, <a href="https://doi.org/10.1007/978-0-387-68361-4_23">New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi)</a>, 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.
%F Lim_{n -> infinity} b(n) = log 4/Pi = 0.24156...
%e a(3) = 3 because b(3) = 1/6 + 0 + 1/21 = 3/14.
%e 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
%o (PARI) a(n) = numerator(-sum(k=1, n, (#binary(k) - 2*hammingweight(k))/(2*k*(2*k+1)))); \\ _Petros Hadjicostas_, May 15 2020
%Y Cf. A037861, A073099, A094640, A110626.
%K easy,frac,nonn
%O 1,3
%A _Jonathan Sondow_, Aug 01 2005
|
