A275322 Decimal expansion of AGM(1, sqrt(2))^2/Pi.

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, 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, 4, 3, 5, 8, 0, 7, 0, 8, 0, 5, 3, 8, 5, 5

Offset: 0

Author

Dimitris Valianatos, Jul 23 2016

Keywords

• nonn
• cons

Comments

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) * ...

Examples

			0.45694658104446362537496662254768...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators, arXiv:2209.04202 [math.CA], 2022, p. 21.

Crossrefs

Cf. A053004 (AGM(1, sqrt(2))).

Cf. A000796, A002161, A068466, A212002.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); 8*Pi(R)^2/Gamma(1/4)^4; // G. C. Greubel, Oct 07 2018

Maple

evalf(GaussAGM(1,sqrt(2))^2/Pi,100); # Muniru A Asiru, Oct 08 2018

Mathematica

First@ RealDigits@ N[ArithmeticGeometricMean[1, Sqrt[2]]^2/Pi, 120] (* Michael De Vlieger, Jul 26 2016 *)

PARI

agm(1, sqrt(2)) ^ 2 / Pi

PARI

8*Pi^2/gamma(1/4)^4 \\ Altug Alkan, Oct 08 2018

Formula

Equals 8*Pi^2/Gamma(1/4)^4 = 4*Gamma(3/4)^2/Gamma(1/4)^2. - Vaclav Kotesovec, Sep 22 2016