

A289090


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


4



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
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OFFSET

1,3


COMMENTS

The pth 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) = ((p1)!!*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 (* JeanFrançois Alcover, Jul 28 2017 *)


PROG

(PARI) \\ General code, for any p > 0:
r(p) = (sqrt(2/Pi)^(p%2)*prod(k=0, (p2)\2, p12*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



