A110894 -id:A110894 - cp's OEIS frontend

A178647 Decimal expansion of the fraction of a population falling within +- 1 standard deviation of the mean, assuming a normal distribution.

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

Offset: 0

Joost de Winter, May 31 2010

			0.6826894921370858971704650912640758449558259334532087819747889004859...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 40, table 40:7:1 at page 387.

Rick L. Shepherd, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Erf.
Wikipedia, 68-95-99.7 rule.

Cf. A110894 (2sigma), A270712 (3sigma).

0.5*(1+erf(1/sqrt(2)))- 0.5*(1+erf(-1/sqrt(2)))

erf(1/sqrt(2)) ; evalf(%) ; # R. J. Mathar, Mar 22 2016

RealDigits[(1 + Erf[1/Sqrt@2])/2 - (1 + Erf[ -1/Sqrt@2])/2, 10, 111][[1]] (* Robert G. Wilson v, Jun 01 2010 *)

(erfc(-1/sqrt(2))-erfc(1/sqrt(2)))/2 \\ Charles R Greathouse IV, Sep 04 2012

1 - erfc(1/sqrt(2)) \\ Rick L. Shepherd, Mar 05 2014

Equals erf(1/sqrt(2)). - Jean-François Alcover, May 29 2013

More terms from Robert G. Wilson v, Jun 01 2010

Edited by N. J. A. Sloane, Jun 07 2010

A270712 Decimal expansion of the fraction of the normal distribution that falls within the 3 sigma error bars.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 22 2016

			0.997300203936739810946696370...

Wikipedia, 68-95-99.7 rule
Wikipedia, Normal distribution

Cf. A178647 (1sigma), A110894 (2sigma).

Maple
```
erf(3/sqrt(2)) ; evalf(%) ;
```

RealDigits[Erf[3/Sqrt[2]], 10, 120][[1]] (* Amiram Eldar, May 24 2023 *)

1 - erfc(3/sqrt(2)) \\ Michel Marcus, Mar 22 2016

A303617 Decimal expansion of Sum_{k >= 0} 2^(2k+1)/Product_{i = 0..k} (2i+1).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Apr 27 2018

			8.83943924091904909456698024436203574171002846378309279604186339401138107...
2/1 + 2^3/(1*3) + 2^5/(1*3*5) + 2^7/(1*3*5*7) + 2^9/(1*3*5*7*9) + 2^11/(1*3*5*7*9*11) + 2^13/(1*3*5*7*9*11*13) + ...

Cf. A001147, A004171.

Cf. A069998, A072334, A110894.

RealDigits[E^2 Sqrt[Pi/2] Erf[Sqrt[2]], 10, 100][[1]]

suminf(k=0, 2^(2*k+1)/prod(i=0, k, (2*i+1))) \\ Michel Marcus, Apr 27 2018

Equals e^2*sqrt(Pi/2)*erf(sqrt(2)) = A072334*A069998*A110894.

A349892 Decimal expansion of erf(1/e).

Original entry on oeis.org

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

Offset: 0

Christoph B. Kassir, Dec 04 2021

			0.3971176949815771696929038299166076792335...

Michael Ian Shamos, Shamos's catalog of the real numbers, (2011).

Cf. A001113, A068985, A099286, A110894.

evalf(erf(exp(-1)), 120);  # Alois P. Heinz, Dec 13 2021

RealDigits[Erf[1/E], 10, 100][[1]] (* Amiram Eldar, Dec 04 2021 *)

1 - erfc(1/exp(1)) \\ Michel Marcus, Dec 04 2021

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A178647 Decimal expansion of the fraction of a population falling within +- 1 standard deviation of the mean, assuming a normal distribution.

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A270712 Decimal expansion of the fraction of the normal distribution that falls within the 3 sigma error bars.

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A303617 Decimal expansion of Sum_{k >= 0} 2^(2*k+1)/Product_{i = 0..k} (2*i+1).

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A349892 Decimal expansion of erf(1/e).

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Author

Keywords

Examples

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Programs

Maple

Mathematica

PARI

A303617 Decimal expansion of Sum_{k >= 0} 2^(2k+1)/Product_{i = 0..k} (2i+1).