OFFSET
1,3
COMMENTS
Entry 34 c of chapter 11 of Ramanujan's second notebook.
This constant is also the ratio T(Pi/2)/T(0), where T(Pi/2) is the exact pendulum period for an amplitude of Pi/2 and T(0) the approximate period 2*Pi*sqrt(L/g) for small angles. - Jean-François Alcover, Aug 05 2014
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..5000
Bruce C. Berndt, Chapter 11 of Ramanujan's second notebook, Bull. Lond. Math. Soc., Vol. 15, No. 4 (1983), 273-320.
Claudio Carvalhaes and Patrick Suppes, Approximations for the period of the simple pendulum based on the arithmetic-geometric mean, American Journal of Physics 76 (2008), 1150-1154.
FORMULA
Equals 2F1([1/2,1/2],[1],1/2) = 1/agm(1, sqrt(1/2)) = gamma(1/4)^2/(2*Pi^(3/2)).
Equals 2*sqrt(2)*K(-1)/Pi, where K is the complete elliptic integral of the first kind, K(-1) being A085565. - Jean-François Alcover, Jun 03 2014
Equals Product_{k>=1} (1-(-1)^k/(2*k)) = 3/2 * 3/4 * 7/6 * 7/8 * 11/10 * 11/12 * ... . - Richard R. Forberg, Dec 05 2015
Reciprocal of A096427. Equals ( Sum_{n = -inf..inf} exp(-Pi*n^2) )^2, a rapidly converging series. For example, summing from n = -5 to n = 5 gives the constant correct to 49 decimal places. - Peter Bala, Mar 06 2019
Equals Sum_{k>=0} binomial(2*k,k)^2/2^(5*k). - Amiram Eldar, Aug 26 2020
Equals (3/2)*hypergeom([-1/4, 3/4], [3/2], 1). - Peter Bala, Mar 04 2022
Equals A175573^2. - Amiram Eldar, Jul 04 2023
EXAMPLE
1.18034059901609622604533794..
MAPLE
sqrt(Pi)/GAMMA(3/4)^2 ; evalf(%) ;
MATHEMATICA
First@ RealDigits[N[Sqrt@ Pi/Gamma[3/4]^2, 120]] (* Michael De Vlieger, Dec 06 2015 *)
PROG
(PARI) sqrt(Pi)/gamma(3/4)^2 \\ Altug Alkan, Dec 05 2015
(MATLAB) sqrt(pi)/gamma(3/4)^2 \\ Altug Alkan, Dec 05 2015
CROSSREFS
KEYWORD
AUTHOR
R. J. Mathar, Jul 15 2010
EXTENSIONS
A-number typo for sqrt(Pi) corrected by R. J. Mathar, Aug 01 2010
STATUS
approved