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A193887
Decimal expansion of Pi * sqrt(2)/8.
7
5, 5, 5, 3, 6, 0, 3, 6, 7, 2, 6, 9, 7, 9, 5, 7, 8, 0, 8, 7, 6, 9, 8, 5, 1, 2, 3, 7, 5, 7, 5, 8, 6, 7, 1, 2, 3, 2, 6, 8, 2, 7, 7, 1, 1, 1, 7, 1, 9, 6, 1, 2, 7, 7, 8, 8, 5, 6, 7, 4, 4, 5, 0, 8, 6, 9, 5, 5, 4, 3, 4, 9, 1, 3, 7, 4, 3, 4, 2, 3, 8, 8, 2, 1, 9, 1, 5, 8, 6, 6, 8, 4, 5, 5, 9, 5, 6, 5, 4, 6, 7, 0, 4, 2, 6
OFFSET
0,1
COMMENTS
This number arises as an addend in one way of giving the closed form of sum(k>=0, (-1)^k/(4*k + 1) ), for example, in Spiegel et al. (2009).
REFERENCES
Stephen Hawking, God Created the Integers: The Mathematical Breakthroughs That Changed History, Running Press, 2007. See p. 573.
Murray R. Spiegel, Seymour Lipschutz, and John Liu, Mathematical Handbook of Formulas and Tables, 3rd Ed. Schaum's Outline Series. New York: McGraw-Hill (2009): p. 135, equation 21.17
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
Piotr Garbaczewski and Vladimir A. Stephanovich, Semigroup modeling of confined Levy flights, arXiv:1106.1530 [cond-mat.stat-mech], 2011, p. 8, equation 40.
Piotr Garbaczewski and Vladimir A. Stephanovich, Dynamics of Confined Lévy Flights in Terms of (Lévy) Semigroups, Acta Phys. Pol. B 43, 977-999 (2012). See p. 992.
FORMULA
Equals Pi/(4*sqrt(2)).
Equals Sum_{k >= 0} (-1)^k * (4*k + 2)/((4*k + 1)*(4*k + 3)). - Peter Bala, Sep 21 2016
From Amiram Eldar, Aug 15 2020: (Start)
Equals Integral_{x=0..oo} 1/(x^2 + 8) dx.
Equals Integral_{x=0..oo} 1/(8*x^2 + 1) dx.
Equals Integral_{x=0..oo} 1/(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7) dx. (End)
Equals Integral_{x=0..Pi/2} sqrt(sin(x)^3*cos(x)) dx. - Stefano Spezia, Apr 29 2026
Equals Integral_{x=0..oo} x^2/(x^4 + 16) dx. - Stefano Spezia, May 03 2026
Equals (1/2)*Sum_{k>0} (-1)^floor((k-1)/2)/(2*k - 1) [Euler, 1748] (see Hawking). - Stefano Spezia, Jul 17 2026
EXAMPLE
0.55536036726979578088...
MATHEMATICA
RealDigits[(Pi Sqrt[2])/8, 10, 100][[1]]
PROG
(PARI) Pi*sqrt(2)/8 \\ G. C. Greubel, Feb 02 2018
(Magma) R:= RealField(); Pi(R)*Sqrt(2)/8; // G. C. Greubel, Feb 02 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Alonso del Arte, Aug 07 2011
STATUS
approved