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%I #36 Dec 05 2020 04:51:26
%S 3,5,8,2,7,8,6,8,3,1,8,5,2,2,0,4,1,7,5,1,5,4,7,0,7,8,5,9,1,5,5,6,1,0,
%T 6,6,6,3,9,2,0,8,5,0,2,3,4,7,5,5,4,8,0,7,7,4,8,0,4,6,2,7,8,4,7,6,8,8,
%U 9,7,3,2,7,9,5,2,6,5,2,5,5,4,7,0,4,2,4,8,5,1,6,1,3,1,7,5,5,2,4,9
%N Decimal expansion of the length of the dipole curve.
%C The Cartesian equation of the dipole curve, also known as the Playfair curve, is (x^2 + y^2)^3 = a^4*x^2, where the parameter 'a' is the area and the width of one lobe. The computation of the length of one lobe is done here with a=1.
%H G. C. Greubel, <a href="/A222494/b222494.txt">Table of n, a(n) for n = 1..5000</a>
%H Robert Ferréol, <a href="http://www.mathcurve.com/courbes2d/dipole/dipole.shtml">Courbe du dipole</a> (in French)
%H Jan Wassenaar, <a href="http://www.2dcurves.com/sextic/sexticdp.html">Dipole curve</a>
%F Equals 2*Integral_{x=0..1} sqrt(1 + f'(x)^2), where f(x) = sqrt(x^(2/3) - x^2).
%F Equals Integral_{t=0..Pi/2} sqrt(3*cos(t)+1/cos(t)). - _Jan Mangaldan_, Nov 23 2020
%e 3.582786831852204175154707859155610666392085023475548077480462784768897327952...
%t 2*a*Sqrt[Pi]*Gamma[5/4]*Hypergeometric2F1[-1/2, 1/4, 3/4, -3]/Gamma[3/4] /. a -> 1 // RealDigits[#, 10, 100] & // First
%t RealDigits[Sqrt[2 Pi] Gamma[5/4] Hypergeometric2F1[1/4, 5/4, 3/4, 3/4]/Gamma[3/4], 10, 100][[1]] (* _Jan Mangaldan_, Nov 22 2020 *)
%o (PARI) localprec(100); intnum(t=0, Pi/2, sqrt(3*cos(t)+1/cos(t))) \\ _Michel Marcus_, Dec 05 2020
%K nonn,cons
%O 1,1
%A _Jean-François Alcover_, May 29 2013
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