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%I #44 Apr 26 2021 06:27:04
%S 1,1,6,1,5,2,7,8,8,9,2,7,4,4,7,7,3,6,7,0,0,4,5,9,5,0,3,1,0,9,6,3,1,7,
%T 9,0,5,5,2,4,8,1,7,3,4,3,0,0,9,6,3,2,3,4,2,4,7,6,9,9,5,5,4,1,4,3,8,2,
%U 2,6,9,1,7,1,6,0,8,7,3,1,9,4,6,0,7,7,5,5,6,1,2,7,2,9,8,3,5,7,1,6,8,4
%N Decimal expansion of a constant appearing in a theorem by Árpád Baricz about Mills' ratio of the standard normal distribution.
%H G. C. Greubel, <a href="/A245967/b245967.txt">Table of n, a(n) for n = 1..5000</a>
%H Árpád Baricz, <a href="https://doi.org/10.1016/j.jmaa.2007.09.063">Mills' ratio: Monotonicity patterns and functional inequalities</a>, Journal of Mathematical Analysis and Applications, vol.340, no.2, Apr 15 2008, pp.1362-1370
%H Iosif Pinelis, <a href="http://www.researchgate.net/publication/235412530">Monotonicity properties of the relative error of a Padé approximation for Mills’ ratio</a>, Journal of Inequalities in Pure and Applied Mathematics 01/2002; 3.
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/MillsRatio.html">Mills ratio</a>
%F The unique positive root of the transcendent equation x*(x^2+2)*r(x) = x^2+1, where r(x) is Mills' ratio exp(x^2/2)*sqrt(Pi/2)*erfc(x/sqrt(2)).
%F r(0) = sqrt(Pi/2).
%F Asymptotic expansion: r(x) ~ 1/x - 1/x^3 + 3/x^5 - 3*5/x^7 + ... + (-1)^k*(2k-1)!!/x^(2k+1) + ...
%e 1.16152788927447736700459503109631790552481734300963234247699554...
%t r[x_] := Exp[x^2/2]*Sqrt[Pi/2]*Erfc[x/Sqrt[2]]; x0 = x /. FindRoot[x*(x^2+2)*r[x] == (x^2+1), {x, 1}, WorkingPrecision -> 102]; RealDigits[x0] // First
%o (PARI) solve(x=1,2,x*(x^2+2)*exp(x^2/2)*sqrt(Pi/2)*erfc(x/sqrt(2))-x^2-1) \\ _Charles R Greathouse IV_, Apr 30 2015
%o (Python)
%o from mpmath import *
%o mp.dps=103
%o print([int(n) for n in list(str(findroot(lambda x: x*(x**2+2)*exp(x**2/2)*sqrt(pi/2)*erfc(x/sqrt(2))-x**2-1, (1, 2))).replace('.', ''))]) # _Indranil Ghosh_, Jul 07 2017
%Y Cf. A001147.
%K nonn,cons,easy
%O 1,3
%A _Jean-François Alcover_, Apr 28 2015
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