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%I #39 Apr 15 2024 04:16:46
%S 2,3,2,7,1,8,5,1,4,2,4,3,6,5,3,8,7,5,0,6,0,5,0,3,6,2,8,5,6,1,8,3,5,7,
%T 0,7,7,5,1,5,1,8,1,7,5,8,2,3,2,5,4,1,1,7,4,7,9,3,2,0,8,1,9,9,4,4,6,1,
%U 1,8,8,2,5,7,3,1,3,6,0,4,9,5,7,8,2,2,5,9,0,0,7,0,1,1,0,6,6,1,0,5,6,2,3,7,1
%N Decimal expansion of the complete elliptic integral of the first kind at sqrt(2*sqrt(2) - 2).
%H G. C. Greubel, <a href="/A262427/b262427.txt">Table of n, a(n) for n = 1..10000</a>
%H M. L. Glasser and V. E. Wood, <a href="http://dx.doi.org/10.1090/S0025-5718-71-99714-6">A closed form evaluation of the elliptic integral</a>, Math. Comp. 25 (1971), 535-536.
%F Equals Pi^(3/2)*sqrt(4 + 2*sqrt(2))/(4*Gamma(5/8)*Gamma(7/8)).
%F Also equals sqrt(2)*K(sqrt(2) - 1).
%F Also equals 2*Integral_{x=0..1} 1/sqrt(1-x^8) dx. - _Christian N. Hofmann_, Jun 24 2023
%F Also equals Pi^(3/2)*cos(Pi/4)*cos(Pi/8)/(Gamma(5/8)*Gamma(7/8)). - _Christian N. Hofmann_, Aug 20 2023
%F Equals Gamma(1/8)^2 / (2^(11/4) * Gamma(1/4)). - _Vaclav Kotesovec_, Apr 15 2024
%e 2.3271851424365387506050362856183570775151817582325411747932...
%p evalf(sqrt(2)*EllipticK(sqrt(2)-1), 120); # _Vaclav Kotesovec_, Sep 22 2015
%p evalf(int(2/sqrt(1-x^8), x=0..1), 120); # _Christian N. Hofmann_, Jun 28 2023
%t K[x_] := EllipticK[x^2/(x^2 - 1)]/Sqrt[1 - x^2]; RealDigits[ K[Sqrt[2 Sqrt[2] - 2]], 10, 105][[1]]
%o (PARI) ellk(k)=intnum(t=0,1,1/sqrt((1-t^2)*(1-(k*t)^2)))
%o sqrt(2)*ellk(sqrt(2)-1) \\ _Charles R Greathouse IV_, Apr 18 2016
%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)^(3/2)*Sqrt(4 + 2*Sqrt(2))/(4*Gamma(5/8)*Gamma(7/8)); // _G. C. Greubel_, Oct 07 2018
%Y Cf. A130786.
%K cons,nonn
%O 1,1
%A _Jean-François Alcover_, Sep 22 2015
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