A099287 -id:A099287 - cp's OEIS frontend

A111129 Decimal expansion of the continued fraction 1+1/(1+2/(1+3/(1+4/(1+5/(1+...))))).

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

Offset: 1

N. J. A. Sloane, based on correspondence from Tom Raes (tommy1729(AT)hotmail.com) and Steven Finch, Sep 22 2005

			1.52513527616098120908909053639057871330711636492060333554631394242...

B. C. Berndt, Y.-S. Choi and S.-Y. Kang, The problems submitted by Ramanujan to the Journal of the Indian Mathematical Society, Continued Fractions: From Analytic Number Theory to Constructive Approximation, ed. B. C. Berndt and F. Gesztesy, Amer. Math. Soc., 1999, pp. 15-56.
S. R. Finch, "Mathematical Constants", Cambridge, pp. 423-428.
H. S. Wall, Analytic Theory of Continued Fractions, Van Nostrand, 1948, pp. 356-358, 367

G. C. Greubel, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Eric Weisstein's World of Mathematics, Generalized Continued Fraction

Cf. A099287, A108088, A111188.

Cf. A225435, A225436 (numerators and denominators of convergents to c.f.).

RealDigits[1/(Sqrt[Pi*E/2]*Erfc[1/Sqrt[2]]), 10, 111][[1]]

1/(sqrt(Pi*exp(1)/2)*erfc(1/sqrt(2))) \\ G. C. Greubel, Jan 24 2017

Equals the reciprocal of sqrt(pi*e/2)*erfc(1/sqrt(2)), where erfc is the complementary error function.

More terms from Robert G. Wilson v and Hans Havermann, Oct 17 2005

Definition corrected by Steven Finch, Feb 05 2009

A099286 Decimal expansion of the error function at 1.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Oct 08 2004

			0.84270079294971486934122063508260925929606699796630290845993789783...

Eric Weisstein's World of Mathematics, Erf

Cf. A007680, A099287, A351400.

Mathematica
```
RealDigits[ Erf[1], 10, 105][[1]]
```
PARI
```
1 - erfc(1)
```

Equals 1-A099287.
Equals (1/e) Sum_{n >= 0} (1/(n/2)!) - 1. - Jean-François Alcover, Jun 14 2020
From Amiram Eldar, Jul 22 2020: (Start)
Equals (2/sqrt(Pi)) * Integral_{x=0..1} exp(-x^2) dx.
Equals (2/sqrt(Pi)) * Sum_{k>=0} (-1)^k/(k! * (2*k + 1)) = (2/sqrt(Pi)) * Sum_{k>=0} (-1)^k/A007680(k).
Equals (1/e) * Sum_{k>=1} 1/Gamma(k + 1/2). (End)

A257526 Decimal expansion of ePierfc(1).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Apr 28 2015

			1.343293421646735170437123594410589778322829567130036872051955564553...

MathOverflow, Contour integration problem from probability
Eric Weisstein's MathWorld, Erfc

Cf. A099287, A108088.

RealDigits[E*Pi*Erfc[1], 10, 103] // First

exp(1)*Pi*erfc(1) \\ Charles R Greathouse IV, Apr 18 2016

Equals Integral_{-infinity..infinity} exp(-x^2)/(1+x^2) dx.
Also equals J(0) where J(c) = Integral_{-infinity..infinity} exp(-(x-c)^2)/(1+x^2) dx = (1/2)*Pi*e*(erfc[1-c*i]*e^(-2*c*i) + erfc[1+c*i]*e^(2*c*i)), where the integrand comes from a shifted normal PDF times a Cauchy PDF.
Equals 2 * Integral_{x=0..Pi/2} exp(-tan(x)^2) dx. - Amiram Eldar, Aug 07 2020

A364618 Decimal expansion of Sum_{k>=0} erfc(k), where erfc(x) is the complementary error function.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jul 30 2023

			1.16199904794712636353230832245579717116634350622586...

Tyma Gaidash, John Barber, and Steven Clark, How to evaluate Sum_{x=0..oo} erfc(x) = 1.1619990479471263635323...?, Mathematics StackExchange, 2021.
Eric Weisstein's World of Mathematics, Dawson's Integral.
Eric Weisstein's World of Mathematics, Erfc.
Wikipedia, Dawson function.
Wikipedia, Error function.

Cf. A099287.

evalf(sum(erfc(k), k = 0 .. infinity), 120)

RealDigits[N[Sum[Erfc[k], {k, 0, Infinity}], 120]][[1]]

PARI
```
sumpos(k = 0, erfc(k))
```

Equals 1 + (2/Pi) * Integral_{x>=1} floor(x) * exp(-x^2) dx.
Equals 1/2 + 1/sqrt(Pi) + (4/sqrt(Pi)) * Sum_{k>=1} D(Pi*k)/(Pi*k), where D(x) is the Dawson function.
Equals (2/Pi)*Integral_{x=0..oo} (exp(x) - cos(x))*sin((x^2)/2)/(x*(cosh(x) - cos(x))) dx. - Velin Yanev, Oct 11 2024

cp's OEIS Frontend

A111129 Decimal expansion of the continued fraction 1+1/(1+2/(1+3/(1+4/(1+5/(1+...))))).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A099286 Decimal expansion of the error function at 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A257526 Decimal expansion of e*Pi*erfc(1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A364618 Decimal expansion of Sum_{k>=0} erfc(k), where erfc(x) is the complementary error function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A257526 Decimal expansion of ePierfc(1).