|
%I #71 Sep 30 2022 23:32:44
%S 4,5,6,9,4,6,5,8,1,0,4,4,4,6,3,6,2,5,3,7,4,9,6,6,6,2,2,5,4,7,6,8,3,3,
%T 3,6,6,1,1,7,6,7,7,3,0,0,1,4,8,3,1,5,0,8,3,9,4,3,6,2,2,4,7,2,6,7,4,8,
%U 4,3,5,8,0,7,0,8,0,5,3,8,5,5
%N Decimal expansion of AGM(1, sqrt(2))^2/Pi.
%C Conjecture: Equals Product_{n odd} (n/(n+2) if n == 1 (mod 4), (n+2)/n otherwise) = (1/3) * (5/3) * (5/7) * (9/7) * (9/11) * (13/11) * (13/15) * (17/15) * (17/19) * (21/19) * (21/23) * (25/23) * (25/27) * ...
%H G. C. Greubel, <a href="/A275322/b275322.txt">Table of n, a(n) for n = 0..10000</a>
%H Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, <a href="https://arxiv.org/abs/2209.04202">The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators</a>, arXiv:2209.04202 [math.CA], 2022, p. 21.
%F Equals 8*Pi^2/Gamma(1/4)^4 = 4*Gamma(3/4)^2/Gamma(1/4)^2. - _Vaclav Kotesovec_, Sep 22 2016
%e 0.45694658104446362537496662254768...
%p evalf(GaussAGM(1,sqrt(2))^2/Pi,100); # _Muniru A Asiru_, Oct 08 2018
%t First@ RealDigits@ N[ArithmeticGeometricMean[1, Sqrt[2]]^2/Pi, 120] (* _Michael De Vlieger_, Jul 26 2016 *)
%o (PARI) agm(1, sqrt(2)) ^ 2 / Pi
%o (PARI) 8*Pi^2/gamma(1/4)^4 \\ _Altug Alkan_, Oct 08 2018
%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); 8*Pi(R)^2/Gamma(1/4)^4; // _G. C. Greubel_, Oct 07 2018
%Y Cf. A053004 (AGM(1, sqrt(2))).
%Y Cf. A000796, A002161, A068466, A212002.
%K nonn,cons
%O 0,1
%A _Dimitris Valianatos_, Jul 23 2016
|
