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

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

1,3

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).

Links

Stanislav Sykora, Table of n, a(n) for n = 1..1000

Wikipedia, Normal distribution

Formula

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

Example

1.16857525496246554867047601109768527106052404816790797238351628742...

Mathematica

ExpectedValue[Abs[#]^3&, NormalDistribution[0, 1]]^(1/3) // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Jul 28 2017 *)

Prog

(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

Crossrefs

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

Sequence in context: A010501 A296426 A249282 * A260691 A296845 A030644

Adjacent sequences: A289087 A289088 A289089 * A289091 A289092 A289093

Keyword

nonn,cons

Author

Stanislav Sykora, Jul 26 2017

Status

approved