A180434 Decimal expansion of constant (2 - Pi/2).

4, 2, 9, 2, 0, 3, 6, 7, 3, 2, 0, 5, 1, 0, 3, 3, 8, 0, 7, 6, 8, 6, 7, 8, 3, 0, 8, 3, 6, 0, 2, 4, 8, 5, 5, 7, 9, 0, 1, 4, 1, 5, 3, 0, 0, 3, 1, 2, 4, 4, 7, 0, 8, 9, 5, 1, 2, 5, 2, 7, 7, 0, 3, 8, 4, 6, 0, 9, 1, 7, 9, 6, 8, 5, 6, 8, 9, 5, 5, 0, 0, 6, 8, 5, 9, 8, 2, 5, 8, 7, 3, 2, 8, 9, 4, 1, 4, 6, 6

Offset: 0

Author

Jonathan Vos Post, Sep 05 2010

Keywords

• cons
• easy
• nonn

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

Examples

			0.42920367320510338076867830836024855790141530...
		

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.
• Index entries for transcendental numbers

Crossrefs

Cf. A004601, A153799, A180433, A222362.

Programs

Mathematica

RealDigits[2-Pi/2,10,120][[1]] (* Harvey P. Dale, Oct 12 2013 *)

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

Extensions

Corrected by Carl R. White, Sep 09 2010

More terms from N. J. A. Sloane, Sep 23 2010