A099286 -id:A099286 - cp's OEIS frontend

A099287 Decimal expansion of the complementary error function at 1.

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

Offset: 0

Robert G. Wilson v, Oct 11 2004

			0.157299207050285130658779364917390740703933002033697091540062102165...

Mathematica
```
RealDigits[ Erfc[1], 10, 105][[1]]
```

1-A099286.

A333419 Decimal expansion of the error function at 1/2, erf(1/2).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Mar 20 2020

			0.5204998778130465376827466538919645287364515757579637...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Wikipedia, Error Function.

Cf. A099286 (erf(1)).

evalf(erf(1/2), 140);  # Alois P. Heinz, Mar 20 2020

RealDigits[Erf[1/2], 10, 120][[1]] (* Amiram Eldar, May 31 2023 *)

1 - erfc(1/2) \\ Michel Marcus, Mar 21 2020

A222392 Decimal expansion of Sum_{n>=1} 1/Gamma(n/2).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Mar 19 2013

			5.57316966431003975325790404977558240053838531356466534303248120641829030...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:5:12 at page 415.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A222391.

Cf. A001113, (Sum_{n>=1} 1/Gamma(n)).

RealDigits[(1 + Erf[1]) E + 1/Sqrt[Pi], 10, 90][[1]]

Equals (1+erf(1))*e+1/sqrt(Pi), where erf is the error function (see A099286).

A351400 Decimal expansion of e * erf(1), where erf is the error function.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Feb 10 2022

The sum of reciprocals of the factorials of the positive half-integers.

			2.29069825230323823094953712686214731693708759053570...

Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, and Sergei Rogosin, Mittag-Leffler Functions, Related Topics and Applications, New York, NY: Springer, 2020. See p. 94, eq. (4.12.9.5).
Constantin Milici, Gheorghe Drăgănescu, and J. Tenreiro Machado, Fractional Differential Equations, Introduction to Fractional Differential Equations, Springer, Cham, 2019. See p. 12, eq. (1.9).

Eric Weisstein's World of Mathematics, Erf.
Eric Weisstein's World of Mathematics, Mittag-Leffler Function.
Wikipedia, Mittag-Leffler function.

Cf. A000079, A001113, A001147, A087197, A099286, A125961, A351401.

evalf(exp(1)*erf(1), 120);  # Alois P. Heinz, Feb 10 2022

Mathematica
```
RealDigits[E * Erf[1], 10, 100][[1]]
```

exp(1)*(1 - erfc(1)) \\ Michel Marcus, Feb 10 2022

Equals Sum_{k>=0} 1/(k + 1/2)! = Sum_{k>=1} 1/Gamma(k + 1/2).
Equals E_{1, 3/2}(1), where E_{a,b}(z) is the two-parameter Mittag-Leffler function.
Equals (1/sqrt(Pi)) * Sum_{k>=1} 2^k/(2*k-1)!! = (1/sqrt(Pi)) * Sum_{k>=1} A000079(k)/A001147(k).
Equals A001113 * A099286.
Equals A087197 * A125961.

A347151 Decimal expansion of erf(2).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Aug 19 2021

			0.995322265018952734162069256367252928610...

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

Cf. A099286.

First[RealDigits[N[Erf[2],87]]] (* Stefano Spezia, Aug 20 2021 *)

1 - erfc(2) \\ Michel Marcus, Aug 20 2021

Equals (2/sqrt(Pi)) * Integral_{x=0..2} exp(-x^2) dx.

Equals (2/sqrt(Pi)) * Sum_{k>=0} (-1)^k * 2^(2*k+1) / (k! * (2*k+1)).

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

A378942 Decimal expansion of (1/sqrt(Pi) + e*erfc(-1))/2.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Dec 11 2024

			2.7865848321550198766289520248877912002691926567823...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 4.6, p. 262.

Cf. A001113, A019739, A087197, A087198, A099286, A222392.

RealDigits[(1/Sqrt[Pi]+E Erfc[-1])/2,10,100][[1]]

Equals (1 + e*sqrt(Pi)*(1 + erf(1)))/(2*sqrt(Pi)).

Equals A222392 / 2. - Amiram Eldar, Feb 15 2025

cp's OEIS Frontend

A099287 Decimal expansion of the complementary error function at 1.

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A333419 Decimal expansion of the error function at 1/2, erf(1/2).

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A222392 Decimal expansion of Sum_{n>=1} 1/Gamma(n/2).

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A351400 Decimal expansion of e * erf(1), where erf is the error function.

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A347151 Decimal expansion of erf(2).

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A349892 Decimal expansion of erf(1/e).

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Maple

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A378942 Decimal expansion of (1/sqrt(Pi) + e*erfc(-1))/2.

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