A126389 - OEIS

%I #10 Feb 16 2025 08:33:04

%S 1,-1,2,-2,-1,1,1,-1,1,-1,3,-3,-2,2,2,-2,2,-2,2,-2,4,-4,-3,3,-1,1,-1,

%T 1,1,-1,-1,1,1,-1,1,-1,3,-3,-1,1,1,-1,1,-1,3,-3,1,-1,3,-3,3,-3,5,-5,

%U -4,4,-2,2,-2,2,-2,2,2,-2,-2,2,2,-2,2,-2,2,-2,4,-4,-2,2

%N Numerators in a series for the "alternating Euler constant" log(4/Pi).

%C Nonzero values of (-1)^n*b(floor(n/2)) for n > 1, where b(n) = (# of 1's) - (# of 0's) in the base 2 expansion of n. The denominators of the series are A126388.

%H J. Sondow, <a href="http://arXiv.org/abs/math.CA/0211148"> 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 J. Sondow, <a href="http://arXiv.org/abs/math.NT/0508042">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.

%H Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/DigitCount.html">Digit Count</a>

%F Log(4/Pi) = 1/2 - 1/3 + 2/6 - 2/7 - 1/8 + 1/9 + 1/10 - 1/11 + 1/12 - 1/13 + 3/14 - 3/15 - 2/16 + 2/17 + 2/22 - ...

%e floor(15/2) = 7 = 111 base 2, which has (# of 1's) - (# of 0's) = 3, so (-1)^15*3 = -3 is a term.

%t b[n_] := DigitCount[n,2,1] - DigitCount[n,2,0]; L = {}; Do[If[b[Floor[n/2]] != 0, L = Append[L,(-1)^n*b[Floor[n/2]]]], {n,2,100}]; L

%Y Cf. A037861, A066879, A094640, A094641, A110625, A110626, A126388.

%K base,sign,changed

%O 2,3

%A _Jonathan Sondow_, Jan 01 2007