A363848 Decimal expansion of the arithmetic mean of the isoperimetric quotient of ellipses when expressed in terms of their eccentricity.

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

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)/4) * Integral_{x=0..1} sqrt(1 - x^2)/E(x)^2 dx.

Example

0.933174653498462644...

Mathematica

First[RealDigits[Pi^2/4 * NIntegrate[Sqrt[1-x^2]/EllipticE[x^2]^2, {x, 0, 1}, WorkingPrecision -> 100]]] (* Stefano Spezia, Jun 24 2023 *)

Crossrefs

Cf. A091476, A363874, A363876.

Sequence in context: A212436 A193015 A021842 * A011112 A138064 A063569

Adjacent sequences: A363845 A363846 A363847 * A363849 A363850 A363851

Keyword

nonn,cons

Author

Tian Vlasic, Jun 24 2023

Extensions

More terms from Stefano Spezia, Jun 24 2023

Status

approved