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A175576 Decimal expansion Pi^(3/2)/Gamma(3/4)^2.

%I A175576 #56 Feb 16 2025 08:33:12
%S A175576 3,7,0,8,1,4,9,3,5,4,6,0,2,7,4,3,8,3,6,8,6,7,7,0,0,6,9,4,3,9,0,5,2,0,
%T A175576 0,9,2,4,3,5,1,9,7,6,4,7,0,4,3,5,3,3,8,1,1,1,7,1,8,5,6,0,9,0,1,1,2,0,
%U A175576 4,3,5,5,3,6,7,6,2,3,9,9,5,6,7,1,4,5,4,3,7,2,3,3,0,0,7,4,3,7,9,4,5,5,5,5,4
%N A175576 Decimal expansion Pi^(3/2)/Gamma(3/4)^2.
%C A175576 Entry 34 e of chapter 11 of Ramanujan's second notebook.
%C A175576 In addition, Pi^(3/2) / Gamma(3/4)^2 is the area of the unit "squircle" as defined in MathWorld. (Note that 8*Gamma(5/4)^2 / sqrt(Pi) is the same constant.) - _Jean-François Alcover_, Feb 24 2011
%C A175576 Real period of the elliptic curve y^2 = x*(x - 1)*(x - 1/2). See Rouse. - _Peter Bala_, Dec 06 2024
%H A175576 G. C. Greubel, <a href="/A175576/b175576.txt">Table of n, a(n) for n = 1..5000</a>
%H A175576 Bruce C. Berndt, <a href="http://dx.doi.org/10.1112/blms/15.4.273">Chapter 11 of Ramanujan's second notebook</a>, Bull. Lond. Math. Soc. vol 15 no 4 (1983) 273-320.
%H A175576 Jeremy Rouse, <a href="http://dx.doi.org/10.1007/s11139-006-0073-3">Hypergeometric functions and elliptic curves</a>, Ramanujan Journal, Vol. 12 (2006), pp. 197-205.
%H A175576 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squircle.html">Squircle</a>
%F A175576 Equals A175476 / A068465^2 = 1/A175575.
%F A175576 Equals Integral_{-oo, oo} 1/(1+2*x^2)^(3/4) or Integral_{-oo, oo} 1/sqrt(1+x^4). - _Jean-François Alcover_, Jun 04 2013
%F A175576 Equals sqrt(2)*L, where L is the lemniscate constant A062539. - _Jean-François Alcover_, Aug 11 2014
%F A175576 From _Peter Bala_, Mar 01 2022 : (Start)
%F A175576 Equals 3*Sum_{n >= 0} (1/(4*n+1) + 1/(4*n-3))*binomial(1/2,n). Cf. A290570.
%F A175576 Equals hypergeom([-1/2, 3/4, -3/4], [-1/4, 5/4], -1).
%F A175576 Equals 2*hypergeom([1/4, 3/4], [5/4], 1) = (16/5)*hypergeom([-1/4, -3/4], [5/4], 1). (End)
%F A175576 Equals 2 * A093341. - _R. J. Mathar_, Dec 08 2023
%F A175576 From _Peter Bala_, Dec 06 2024: (Start)
%F A175576 Equals Pi*hypergeom([1/2, 1/2], [1], 1/2).
%F A175576 Equals 2*Integral_{x = 0..Pi/2} 1/sqrt(1 - (1/2)*sin^2(x)) dx. See Rouse. (End)
%e A175576 3.708149354602743836867700694390520092435197647...
%p A175576 Pi^(3/2)/GAMMA(3/4)^2 ; evalf(%) ;
%t A175576 RealDigits[Pi*EllipticTheta[3, 0, Exp[-Pi]]^2, 10, 50][[1]]
%t A175576 RealDigits[Pi^(3/2)/(Gamma[3/4])^2, 10, 50][[1]] (* _G. C. Greubel_, Feb 12 2017 *)
%o A175576 (PARI) Pi^1.5/gamma(3/4)^2 \\ _Charles R Greathouse IV_, Jun 06 2016
%Y A175576 Cf. A062539, A068465, A175476, A175575, A290570.
%K A175576 nonn,cons,easy
%O A175576 1,1
%A A175576 _R. J. Mathar_, Jul 15 2010