A094641 - OEIS

%I #15 Aug 07 2024 14:45:03

%S 0,4,7,6,3,1,1,9,1,1,4,26,1,2,4,1,9,1,20,3,1,12,1,2,7,1,5,2,1,5,3,1,1,

%T 1,4,1,1,57,1,2,1,8,8,1,1,1,1,1,22,1,1,6,1,6,6,1,3,1,4,2,2,2,4,1,1,2,

%U 1,19,17,348,1,1,5,16,2,2,5,1,5,2,4,2,5,1,11,1,1,11,13,2,1,1,5,2,1,2,10,1,2

%N Continued fraction for the "alternating Euler constant" log(4/Pi).

%C See the Comments in A094640 for why log(4/Pi) is an "alternating Euler constant."

%D G. Boros and V. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, Chap. 7.

%D J. Borwein and P. Borwein, Pi and the AGM, John Wiley & Sons, New York, 1987, Chap. 11.

%H D. 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 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 J. Sondow and P. 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.

%e log(4/Pi) = 0 + 1/(4 + 1/(7 + 1/(6 + 1/(3 + 1/(1 + ...)))))

%t ContinuedFraction[ Log[4/Pi], 100]

%Y Cf. A094640 (decimal expansion of log(4/Pi)).

%K cofr,easy,nonn

%O 0,2

%A _Jonathan Sondow_ and _Robert G. Wilson v_, May 18 2004

%E Offset changed by _Andrew Howroyd_, Aug 07 2024