%I #45 Mar 23 2024 13:50:47

%S 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,

%T 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,

%U 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

%N Decimal expansion of constant (2 - Pi/2).

%C (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

%H Vincenzo Librandi, <a href="/A180434/b180434.txt">Table of n, a(n) for n = 0..1000</a>

%H Robert Ferréol, <a href="https://www.mathcurve.com/courbes2d.gb/strophoiddroite/strophoiddroite.shtml">Right strophoid</a>, Math Curve.

%H Nikita Kalinin and Mikhail Shkolnikov, <a href="https://arxiv.org/abs/1701.07584">The number Pi and summation by SL(2,Z)</a>, arXiv:1701.07584 [math.NT], 2016. Gives a formula.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals Integral_{t=0..Pi/4} ((cos(2*t))/cos(t))^2 dt. - _Bernard Schott_, Jan 28 2020

%F From _Amiram Eldar_, May 30 2021: (Start)

%F Equals Sum_{k>=1} 2^k/(binomial(2*k,k)*k*(2*k + 1)).

%F Equals Integral_{x=0..1} arcsin(x)*arccos(x) dx. (End)

%F Equals Integral_{x=0..1} sqrt(x)/(1+x) dx. - _Andy Nicol_, Mar 23 2024

%F Equals A153799/2. - _Hugo Pfoertner_, Mar 23 2024

%e 0.42920367320510338076867830836024855790141530...

%t RealDigits[2-Pi/2,10,120][[1]] (* _Harvey P. Dale_, Oct 12 2013 *)

%Y Cf. A004601, A153799, A180433, A222362.

%K cons,easy,nonn

%O 0,1

%A _Jonathan Vos Post_, Sep 05 2010

%E Corrected by _Carl R. White_, Sep 09 2010

%E More terms from _N. J. A. Sloane_, Sep 23 2010