cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A011002 Decimal expansion of 4th root of 3.

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

Comments

Also 8th root of 9. - Alonso del Arte, Dec 06 2011
The 4th root r(4) of the expected value E(x^4) for a normal distribution with zero mean and standard deviation 1. See A289090 for more details. - Stanislav Sykora, Jul 26 2017

Examples

			1.3160740129524924608192189...

Links

Crossrefs

Cf. A179615, continued fraction.
Cf. A076668, A011350, A289090, A289091.

Programs

Formula

Equals exp(arctanh(1/2)/2). - Amiram Eldar, Jul 10 2023

A289090 Decimal expansion of (E(|x|^3))^(1/3), with x being a normally distributed random variable.

Original entry on oeis.org

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

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Author

Stanislav Sykora, Jul 26 2017

Keywords

Comments

The p-th root r(p) of the expected value E(|x|^p) for various distributions appears, for example, in chemical physics, where some interactions depend on high powers of interatomic distances.
When x is distributed normally with zero mean and standard deviation 1, r(p) evaluates to r(p) = ((p-1)!!*w(p))^(1/p), where w(p) = 1 for even p and sqrt(2/Pi) for odd p. Note that, by definition, r(2) = 1 and r(1) = w(1) = A076668.
The present constant is a = r(3).

Examples

			1.16857525496246554867047601109768527106052404816790797238351628742...

Links

Crossrefs

Cf. A060294, A076668 (p=1), A011002 (p=4), A289091 (p=5), A011350 (p=6).

Programs

  • Mathematica
    ExpectedValue[Abs[#]^3&, NormalDistribution[0, 1]]^(1/3) // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Jul 28 2017 *)
  • PARI
    \\ General code, for any p > 0:
r(p) = (sqrt(2/Pi)^(p%2)*prod(k=0,(p-2)\2,p-1-2*k))^(1/p);
a = r(3) \\ Present instance

Formula

Equals (2!!*sqrt(2/Pi))^(1/3) = (2*A076668)^(1/3).

A289091 Decimal expansion of (E(|x|^5))^(1/5), with x being a normally distributed random variable.

Original entry on oeis.org

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

Views

Author

Stanislav Sykora, Jul 26 2017

Keywords

Comments

The 5th root r(5) of the expected value E(|x|^5) for a normal distribution with zero mean and standard deviation 1. See A289090 for more details.

Examples

			1.44879190154930528525354659881281058821340103935196780729503058015...

Links

Crossrefs

Cf. A060294, A076668 (p=1), A289090 (p=3), A011002 (p=4), A011350 (p=6).

Programs

  • PARI
    // General code, for any p > 0:
r(p) = (sqrt(2/Pi)^(p%2)*prod(k=0,(p-2)\2,p-1-2*k))^(1/p);
a = r(5) // Present instance

Formula

a = r(5), where r(p) = ((p-1)!!*sqrt(2/Pi))^(1/p).
a = (8*A076668)^(1/5).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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