A094640 Decimal expansion of the "alternating Euler constant" log(4/Pi).

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

Offset: 0

Author

Jonathan Sondow and Robert G. Wilson v, May 18 2004

Keywords

• cons
• easy
• nonn

Comments

Decimal expansion of Sum_{n>=1} (-1)^{n-1} (1/n - log(1 + 1/n)) (see Sondow 2005), so in comparison to A001620's sum formula, log(4/Pi) is an "alternating Euler constant."

Examples

			log(4/Pi) = 0.24156447527...
		

References

• George Boros and Victor Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, Chap. 7.
• Jonathan Borwein and Peter Borwein, Pi and the AGM, John Wiley & Sons, New York, 1987, Chap. 11.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.
• Dirk Huylebrouck, Similarities in irrationality proofs for Pi, ln2, zeta(2) and zeta(3), arXiv:math/0211148 [math.CA], 2002-2004; Amer. Math. Monthly 108 (2001), 222-231.
• 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; 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; 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.
• Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006; J. Math. Anal. Appl. 332(1) (2007), 292-314.
• Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.
• Eric Weisstein's World of Mathematics, Hadjicostas's Formula.
• Eric Weisstein's World of Mathematics, Digit Count.

Crossrefs

Cf. A094641, A103130, A110625, A110626.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); Log(4/Pi(R)); // G. C. Greubel, Aug 28 2018

Mathematica

RealDigits[ Log[4/Pi], 10, 111][[1]]

PARI

log(4/Pi) \\ Charles R Greathouse IV, Jun 06 2011

Formula

Equals Integral_{x=0..1, y=0..1} (x-1)/((1+x*y)*log(x*y)). (see Sondow 2005).

Equals -Integral_{x=0..1} (1-x)^2 dx/((1+x^2)*log(x)). - Amiram Eldar, Jun 29 2020

From Petros Hadjicostas, Jun 29 2020: (Start)

Equals Integral_{x=0..1} (1 - x + log(x))/((1 + x)*log(x)) dx. (Let u = x*y and v = y in Sondow's double integral and integrate w.r.t. v.)

Equals Integral_{x=0..1, y=0..1} (1 - x*y)^2/((1 + x^2*y^2)*(log(x*y))^2). (Apply Glasser's (2019) Theorem 1 on Amiram Eldar's integral above.) (End)

Equals Integral_{0..Pi/2} (sec(t)-2/(Pi-2*t)) dt. - Clark Kimberling, Jul 10 2020

Equals -Sum_{k>=1} log(1 - 1/(2*k+1)^2). - Amiram Eldar, Jul 06 2023