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A245967 Decimal expansion of a constant appearing in a theorem by Árpád Baricz about Mills' ratio of the standard normal distribution. 1
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

Árpád Baricz, Mills' ratio: Monotonicity patterns and functional inequalities, Journal of Mathematical Analysis and Applications, vol.340, no.2, Apr 15 2008, pp.1362-1370

Iosif Pinelis, Monotonicity properties of the relative error of a Padé approximation for Mills’ ratio, Journal of Inequalities in Pure and Applied Mathematics 01/2002; 3.

Eric Weisstein's MathWorld, Mills ratio

FORMULA

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)).

r(0) = sqrt(Pi/2).

Asymptotic expansion: r(x) ~ 1/x - 1/x^3 + 3/x^5 - 3*5/x^7 + ... + (-1)^k*(2k-1)!!/x^(2k+1) + ...

EXAMPLE

1.16152788927447736700459503109631790552481734300963234247699554...

MATHEMATICA

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

PROG

(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

(Python)

from mpmath import *

mp.dps=103

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

CROSSREFS

Cf. A001147.

Sequence in context: A160135 A178645 A010137 * A212006 A245725 A011096

Adjacent sequences:  A245964 A245965 A245966 * A245968 A245969 A245970

KEYWORD

nonn,cons,easy

AUTHOR

Jean-François Alcover, Apr 28 2015

STATUS

approved

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Last modified July 24 16:24 EDT 2021. Contains 346273 sequences. (Running on oeis4.)