A118178 Decimal expansion of arc length of eight curve.

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

Offset

1,1

Links

Table of n, a(n) for n=1..105.

Eric Weisstein's World of Mathematics, Eight Curve

Formula

4*Integral_{t=0..Pi/2} (sqrt(4*sin(t)^4 - 5*sin(t)^2 + 2)) dt.

Can be expressed in terms of complete elliptic integrals. Using Mathematica notation, with m = (4 + Sqrt[2])/8, the arc length is 4*2^(1/4)*(EllipticE[m] - EllipticK[m]) + (3 + 2*Sqrt[2])*2^(-1/4)*EllipticPi[(4 - 3*Sqrt[2])/8, m]. - David W. Cantrell, Apr 22 2006

Example

6.097223470104916046...

Mathematica

RealDigits[4*NIntegrate[Sqrt[4*Sin[t]^4-5*Sin[t]^2+2], {t, 0, Pi/2}, WorkingPrecision->200], 10, 110][[1]]

Crossrefs

Sequence in context: A189037 A153201 A248725 * A021168 A147709 A176403

Adjacent sequences: A118175 A118176 A118177 * A118179 A118180 A118181

Keyword

nonn,cons,changed

Author

Eric W. Weisstein, Apr 13 2006

Status

approved