A344716 Decimal expansion of (gamma + log(4/Pi))/2, where gamma is Euler's constant.

4, 0, 9, 3, 9, 0, 0, 7, 0, 0, 8, 6, 0, 1, 1, 6, 5, 2, 6, 4, 8, 7, 7, 4, 4, 9, 0, 8, 2, 2, 8, 4, 8, 4, 2, 7, 7, 7, 2, 9, 3, 2, 3, 9, 5, 8, 7, 2, 5, 6, 1, 2, 6, 7, 7, 6, 6, 7, 5, 2, 0, 9, 1, 1, 9, 9, 7, 5, 8, 6, 0, 0, 4, 1, 6, 1, 1, 4, 0, 1, 1, 1, 8, 2, 5, 2, 5, 2, 2, 3, 5, 0, 4, 5, 4, 7, 2, 0, 8, 4, 4, 8, 3, 1, 2

Offset: 0

Author

Kevin Ryde, May 27 2021

Keywords

• nonn
• cons

Examples

			0.40939007008601165264877449082284842...
		

Links

• Jean-Paul Allouche, Jeffrey Shallit, and Jonathan Sondow, Summation of Series Defined by Counting Blocks of Digits, Journal of Number Theory, volume 123, number 1, March 2007, pages 133-143. Also arXiv:math/0512399 [math.NT], 2005-2006.

Crossrefs

Cf. A001620, A094640, A000120.

Programs

Mathematica

RealDigits[(EulerGamma + Log[4/Pi])/2, 10, 100][[1]] (* Amiram Eldar, May 27 2021 *)

Formula

Equals (A001620 + A094640)/2, the mean of Euler's constant and alternating Euler's constant.

Equals Sum_{n>=1} A000120(n) / (2*n*(2*n+1)), where A000120 is the number of 1-bits of n in binary. [Allouche, Shallit, Sondow]

Equals Sum_{k>=1} (1/(2*k-1) - log(1+1/(2*k-1))). - Amiram Eldar, Jun 19 2023