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%I #62 Jul 06 2023 01:53:10
%S 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,
%T 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,
%U 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
%N Decimal expansion of the "alternating Euler constant" log(4/Pi).
%C 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."
%D George Boros and Victor Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, Chap. 7.
%D Jonathan Borwein and Peter Borwein, Pi and the AGM, John Wiley & Sons, New York, 1987, Chap. 11.
%H G. C. Greubel, <a href="/A094640/b094640.txt">Table of n, a(n) for n = 0..10000</a>
%H M. L. Glasser, <a href="https://doi.org/10.1080/00029890.2019.1565856">A note on Beukers's and related integrals</a>, Amer. Math. Monthly 126(4) (2019), 361-363.
%H Dirk Huylebrouck, <a href="https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Huylebrouck222-231.pdf">Similarities in irrationality proofs for Pi, ln2, zeta(2) and zeta(3)</a>, Amer. Math. Monthly 108 (2001), 222-231.
%H Jonathan 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 Jonathan 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 Jonathan Sondow and Petros Hadjicostas, <a href="http://arXiv.org/abs/math/0610499">The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant</a>, J. Math. Anal. Appl. 332(1) (2007), 292-314.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Euler-MascheroniConstant.html">Euler-Mascheroni Constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HadjicostassFormula.html">Hadjicostas's Formula</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigitCount.html">Digit Count</a>.
%F Integral_{x=0..1, y=0..1} (x-1)/((1+x*y)*log(x*y)) - (see Sondow 2005).
%F Equals -Integral_{x=0..1} (1-x)^2 dx/((1+x^2)*log(x)). - _Amiram Eldar_, Jun 29 2020
%F From _Petros Hadjicostas_, Jun 29 2020: (Start)
%F 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.)
%F 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)
%F Equals Integral_{0..Pi/2} (sec(t)-2/(Pi-2*t)) dt. - _Clark Kimberling_, Jul 10 2020
%F Equals -Sum_{k>=1} log(1 - 1/(2*k+1)^2). - _Amiram Eldar_, Jul 06 2023
%e log(4/Pi) = 0.24156447527...
%t RealDigits[ Log[4/Pi], 10, 111][[1]]
%o (PARI) log(4/Pi) \\ _Charles R Greathouse IV_, Jun 06, 2011
%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Log(4/Pi(R)); // _G. C. Greubel_, Aug 28 2018
%Y Cf. A094641, A103130, A110625, A110626.
%K cons,easy,nonn
%O 0,1
%A _Jonathan Sondow_ and _Robert G. Wilson v_, May 18 2004
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