A222494 Decimal expansion of the length of the dipole curve.

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

Offset

1,1

Comments

The Cartesian equation of the dipole curve, also known as the Playfair curve, is (x^2 + y^2)^3 = a^4*x^2, where the parameter 'a' is the area and the width of one lobe. The computation of the length of one lobe is done here with a=1.

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Robert Ferréol, Courbe du dipole (in French)

Jan Wassenaar, Dipole curve

Formula

Equals 2*Integral_{x=0..1} sqrt(1 + f'(x)^2), where f(x) = sqrt(x^(2/3) - x^2).

Equals Integral_{t=0..Pi/2} sqrt(3*cos(t)+1/cos(t)). - Jan Mangaldan, Nov 23 2020

Example

3.582786831852204175154707859155610666392085023475548077480462784768897327952...

Mathematica

2*a*Sqrt[Pi]*Gamma[5/4]*Hypergeometric2F1[-1/2, 1/4, 3/4, -3]/Gamma[3/4] /. a -> 1 // RealDigits[#, 10, 100] & // First
RealDigits[Sqrt[2 Pi] Gamma[5/4] Hypergeometric2F1[1/4, 5/4, 3/4, 3/4]/Gamma[3/4], 10, 100][[1]] (* Jan Mangaldan, Nov 22 2020 *)

Prog

(PARI) localprec(100); intnum(t=0, Pi/2, sqrt(3*cos(t)+1/cos(t))) \\ Michel Marcus, Dec 05 2020

Crossrefs

Sequence in context: A189964 A181910 A367453 * A336238 A181918 A010617

Adjacent sequences: A222491 A222492 A222493 * A222495 A222496 A222497

Keyword

nonn,cons

Author

Jean-François Alcover, May 29 2013

Status

approved