OFFSET
0,1
COMMENTS
(2-Pi/2)*a^2 is the area of the loop of the right strophoid (also called the Newton strophoid) whose polar equation is r = a*cos(2*t)/cos(t) and whose Cartesian equation is x*(x^2+y^2) = a*(x^2-y^2) or y = +- x*sqrt((a-x)/(a+x)). See the curve with its loop at the Mathcurve link; the loop appears for -Pi/4 <= t <= Pi/4. - Bernard Schott, Jan 28 2020
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Robert Ferréol, Right strophoid, Math Curve.
Nikita Kalinin and Mikhail Shkolnikov, The number Pi and summation by SL(2,Z), arXiv:1701.07584 [math.NT], 2016. Gives a formula.
FORMULA
Equals Integral_{t=0..Pi/4} ((cos(2*t))/cos(t))^2 dt. - Bernard Schott, Jan 28 2020
From Amiram Eldar, May 30 2021: (Start)
Equals Sum_{k>=1} 2^k/(binomial(2*k,k)*k*(2*k + 1)).
Equals Integral_{x=0..1} arcsin(x)*arccos(x) dx. (End)
Equals Integral_{x=0..1} sqrt(x)/(1+x) dx. - Andy Nicol, Mar 23 2024
Equals A153799/2. - Hugo Pfoertner, Mar 23 2024
EXAMPLE
0.42920367320510338076867830836024855790141530...
MATHEMATICA
RealDigits[2-Pi/2, 10, 120][[1]] (* Harvey P. Dale, Oct 12 2013 *)
CROSSREFS
KEYWORD
AUTHOR
Jonathan Vos Post, Sep 05 2010
EXTENSIONS
Corrected by Carl R. White, Sep 09 2010
More terms from N. J. A. Sloane, Sep 23 2010
STATUS
approved