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%I #37 Sep 28 2022 14:04:19
%S 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,
%T 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,
%U 4,3,4,9,1,3,7,4
%N Decimal expansion of Pi * sqrt(2)/8.
%C 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).
%D Murray R. Spiegel, Seymour Lipschutz, John Liu. Mathematical Handbook of Formulas and Tables, 3rd Ed. Schaum's Outline Series. New York: McGraw-Hill (2009): p. 135, equation 21.17
%H G. C. Greubel, <a href="/A193887/b193887.txt">Table of n, a(n) for n = 0..10000</a>
%H Piotr Garbaczewski and Vladimir Stephanovich, <a href="http://arxiv.org/abs/1106.1530">Semigroup modeling of confined Levy flights</a>, arXiv:1106.1530 [cond-mat.stat-mech], 2011, p. 8, equation 40.
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F Equals Pi/(4*sqrt(2)).
%F Equals Sum_{k >= 0} (-1)^k * (4*k + 2)/((4*k + 1)*(4*k + 3)). - _Peter Bala_, Sep 21 2016
%F From _Amiram Eldar_, Aug 15 2020: (Start)
%F Equals Integral_{x=0..oo} 1/(x^2 + 8) dx.
%F Equals Integral_{x=0..oo} 1/(8*x^2 + 1) dx.
%F Equals Integral_{x=0..oo} 1/(1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7) dx. (End)
%e 0.55536036726979578088...
%t RealDigits[(Pi Sqrt[2])/8, 10, 100][[1]]
%o (PARI) Pi*sqrt(2)/8 \\ _G. C. Greubel_, Feb 02 2018
%o (Magma) R:= RealField(); Pi(R)*Sqrt(2)/8; // _G. C. Greubel_, Feb 02 2018
%Y Cf. A181048, A063448, A247719, A093954, A244976.
%K nonn,cons
%O 0,1
%A _Alonso del Arte_, Aug 07 2011
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