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A094640
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Decimal expansion of the "alternating Euler constant" log(4/Pi).
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9
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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
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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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."
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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.
D. Huylebrouck, Similarities in irrationality proofs for Pi, ln2, zeta(2) and zeta(3), Amer. Math. Monthly 108 (2001) 222-231.
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LINKS
| J. Sondow, Double Integrals for Euler's Constant and ln(4/Pi) and an analog of Hadjicostas's formula, Amer. Math. Monthly 112 (2005), 61-65.
J. Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), 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.
J. Sondow and P. Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, 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
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FORMULA
| Integral_{x=0..1, y=0..1} (x-1)/((1+x*y)*log(x*y)) - (see Sondow 2005).
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EXAMPLE
| log(4/Pi) = 0.24156447527...
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MATHEMATICA
| RealDigits[ Log[4/Pi], 10, 111][[1]]
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PROG
| (PARI) log(4/Pi) \\ Charles R Greathouse IV, Jun 06, 2011
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CROSSREFS
| Cf. A094641, A103130, A110625, A110626.
Sequence in context: A060370 A165064 A021418 * A070937 A175036 A177985
Adjacent sequences: A094637 A094638 A094639 * A094641 A094642 A094643
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KEYWORD
| cons,easy,nonn
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AUTHOR
| Jonathan Sondow (jsondow(AT)alumni.princeton.edu) and Robert G. Wilson v (rgwv(AT)rgwv.com), May 18 2004
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