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%I #36 Nov 10 2023 12:47:42
%S 2,2,2,1,4,4,1,4,6,9,0,7,9,1,8,3,1,2,3,5,0,7,9,4,0,4,9,5,0,3,0,3,4,6,
%T 8,4,9,3,0,7,3,1,0,8,4,4,6,8,7,8,4,5,1,1,1,5,4,2,6,9,7,8,0,3,4,7,8,2,
%U 1,7,3,9,6,5,4,9,7,3,6,9,5,5,2,8,7,6,6,3,4,6,7,3,8,2,3,8,2,6,1,8,6,8,1,7
%N Decimal expansion of Integral_{t=0..Pi/2} sqrt(tan(t)) dt.
%H G. C. Greubel, <a href="/A247719/b247719.txt">Table of n, a(n) for n = 1..10000</a>
%H D. H. Bailey and J. M. Borwein, <a href="https://escholarship.org/uc/item/4281090t">Highly Parallel, High-Precision Numerical Integration</a> p. 7. (2005) Lawrence Berkeley National Laboratory.
%H Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, pp. 235-236.
%F Equals Pi/sqrt(2).
%F Equals A063448/2.
%F c = 2*( Sum_{k >= 0} (-1)^k/(4*k + 1) + Sum_{k >= 0} (-1)^k/(4*k + 3) ) = 2*(A181048 + A181049). - _Peter Bala_, Sep 21 2016
%F From _Amiram Eldar_, Aug 07 2020: (Start)
%F Equals Integral_{x=0..Pi} 1/(cos(x)^2 + 1) dx = Integral_{x=0..Pi} 1/(sin(x)^2 + 1) dx.
%F Equals Integral_{x=-oo..oo} 1/(x^4 + 1) dx.
%F Equals Integral_{x=-oo..oo} x^2/(x^4 + 1) dx.
%F Equals Integral_{x=0..oo} log(1 + 1/(2 * x^2)) dx. (End)
%F Equals Integral_{x=0..2*Pi} 1/(3 + sin(x)) dx; since for a>1: Integral_{x=0..2*Pi} 1/(a + sin(x)) dx = 2*Pi/sqrt(a^2-1). - _Bernard Schott_, Feb 19 2023
%F Equals 20/9 - 160*Sum_{n >= 1} 1/((64*n^2 - 1)*(64*n^2 - 4)*(64*n^2 - 9)). - _Peter Bala_, Nov 09 2023
%e 2.22144146907918312350794049503034684930731...
%t RealDigits[Pi/Sqrt[2], 10, 104] // First
%o (PARI) default(realprecision, 100); Pi/sqrt(2) \\ _G. C. Greubel_, Sep 07 2018
%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)/Sqrt(2); // _G. C. Greubel_, Sep 07 2018
%Y Cf. A063448, A093954, A193887, A244976, A181048, A181049, A186706.
%K nonn,cons,easy
%O 1,1
%A _Jean-François Alcover_, Sep 23 2014
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