A126389 Numerators in a series for the "alternating Euler constant" log(4/Pi).

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, 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, -4, 4, -2, 2, -2, 2, -2, 2, 2, -2, -2, 2, 2, -2, 2, -2, 2, -2, 4, -4, -2, 2

Offset: 2

Author

Jonathan Sondow, Jan 01 2007

Keywords

• base
• sign

Comments

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.

Examples

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

Links

• J. 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.
• J. 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.
• Eric Weisstein's MathWorld, Digit Count

Crossrefs

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

Programs

Mathematica

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

Formula

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 - ...