A094641 Continued fraction for the "alternating Euler constant" log(4/Pi).

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, 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, 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

Offset: 0

Author

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

Keywords

• cofr
• easy
• nonn

Comments

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

Examples

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

References

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

Links

• D. Huylebrouck, Similarities in irrationality proofs for Pi, ln2, zeta(2) and zeta(3), 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.

Crossrefs

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

Programs

Mathematica

ContinuedFraction[ Log[4/Pi], 100]

Extensions

Offset changed by Andrew Howroyd, Aug 07 2024