|
%I #42 Jul 21 2021 12:23:46
%S 1,9,5,9,1,3,2,7,6,0,1,5,3,0,3,6,3,5,0,8,5,4,2,7,7,7,9,6,1,1,2,1,5,4,
%T 5,5,6,5,8,3,1,4,3,2,4,7,1,9,7,0,0,1,4,3,6,4,3,9,6,9,1,4,9,0,1,1,0,0,
%U 0,7,0,4,3,3,8,7,2,1,1,2,3,4,3,5,1,0,5,9
%N Decimal expansion of arcsec(3)/(2*Pi).
%H S. R. Finch, <a href="http://arxiv.org/abs/1111.4976">Mean width of a regular simplex</a>, arxiv:1111.4976 [math.MG], 2011-2016, S_2.
%H W. V. Gehrlein, <a href="https://www.researchgate.net/publication/257651659_Condorcet%27s_Paradox_and_the_Condorcet_Efficiency_of_Voting_Rules">Condorcet's paradox and the Condorcet efficiency of voting rules</a>, Mathematica Japonica 45, 173-199, 1997.
%F Equals A137914*A086201.
%F From _Robert FERREOL_, Mar 21 2018: (Start)
%F Equals arctan(2*sqrt(2))/(2*Pi).
%F Equals (1/(2*Pi))*Integral_{t>=sqrt(2)/4} 1/(1+t^2).
%F Equals Probability(X>sqrt(2)/4)/2, if X is a Cauchy distributed random variable of location parameter 0 and scale parameter 1.
%F Equals the asymptotic probability p that A is predominantly preferred to B and B predominantly preferred to C when n persons provide a preference list of three candidates A, B, C (with a uniform distribution on voter preferences); the asymptotic probability that A > B > C > A or A > C > B > A (where ">" means "predominantly preferred to") is 3p-1/2 = 8.77...% (Condorcet paradox); the contrary probability (existence of a Condorcet winner) is 3/2-3p = 91.226...%.
%F See Gehrlein link. (End)
%e 0.195913276015303635085427779611215...
%p arcsec(3)/2/Pi ; evalf(%) ;
%t RealDigits[ArcSec[3]/(2 Pi),10,120][[1]] (* _Harvey P. Dale_, Jul 21 2021 *)
%o (Python)
%o from mpmath import mp, asec, pi
%o mp.dps=89
%o print([int(z) for z in list(str(asec(3)/(2*pi))[2:-1])]) # _Indranil Ghosh_, Jul 07 2017
%o (PARI) acos(1/3)/(2*Pi) \\ _Michel Marcus_, Jul 07 2017
%Y Cf. A086201, A137914.
%K nonn,cons,easy
%O 0,2
%A _R. J. Mathar_, Jul 07 2017
|
