A363876 Decimal expansion of the geometric mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

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

Offset

0,1

Comments

The isoperimetric quotient of a curve is defined as Q = (4*Pi*A)/p^2, where A and p are the area and the perimeter of that curve respectively.

The isoperimetric quotient of an ellipse depends only on its eccentricity e in accordance to the formula Q = (Pi^2*sqrt(1-e^2))/(4*E(e)^2), where E() is the complete elliptic integral of the second kind.

Links

Table of n, a(n) for n=0..87.

Eric Weisstein's World of Mathematics, Isoperimetric Quotient

Wikipedia, Elliptic integral

Formula

Equals ((Pi^2)/2) * exp(-1-2*Integral_{x=0..1} log(E(x)) dx).

Example

0.916816923382168248...

Mathematica

First[RealDigits[Pi^2/2*Exp[-1 - 2*NIntegrate[Log[EllipticE[x^2]], {x, 0, 1}, WorkingPrecision -> 100]]]]

Crossrefs

Cf. A102753, A363848, A363874.

Sequence in context: A154902 A160042 A264934 * A195712 A296473 A213916

Adjacent sequences: A363873 A363874 A363875 * A363877 A363878 A363879

Keyword

nonn,cons

Author

Tian Vlasic, Jun 25 2023

Status

approved