A275373 Decimal expansion of the volume of the convex hull of two orthogonal disks in the "two-circle roller" configuration.

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

Offset

1,1

Links

Table of n, a(n) for n=1..101.

Steven R. Finch, Convex Hull of Two Orthogonal Disks, arXiv:1211.4514v3 [math.MG] 2016 p. 15.

Formula

omega = arcsin(sqrt(sqrt(2) - 1)),

gamma = (sqrt(2)-1)/2+E(omega, -1)+sqrt(2)(EllipticPi(-sqrt(2)-1,omega,-1) + EllipticPi(sqrt(2)-1,omega,-1)-F(omega,-1)),

VL = (8*gamma)/(3*sqrt(2)), where E is the complete elliptic integral of the second kind, EllipticPi is the incomplete elliptic integral of the third kind and F is the elliptic integral of the first kind.

Example

3.28181948744968941903219331047186568800423705235770280145947014746...

Mathematica

omega = ArcSin[Sqrt[Sqrt[2] - 1]];
gamma =  (Sqrt[2] - 1)/2 + EllipticE[omega, -1] + Sqrt[2] (EllipticPi[ -Sqrt[2] - 1, omega, -1] + EllipticPi[Sqrt[2] - 1, omega, -1] - EllipticF[omega, -1]);
VL = (8*gamma)/(3*Sqrt[2]);
RealDigits[VL, 10, 101][[1]]
RealDigits[2/3 (4 EllipticE[ArcTan[2^(1/4)], 1/2] - 2 Sqrt[2] EllipticF[ArcTan[2^(1/4)], 1/2] + 2 (Sqrt[2] + 1) EllipticPi[-1/Sqrt[2], ArcTan[2^(1/4)], 1/2] + 2 (Sqrt[2] - 1) EllipticPi[1/Sqrt[2], ArcTan[2^(1/4)], 1/2] + Sqrt[2] - 2), 10, 101][[1]] (* Jan Mangaldan, Jan 04 2017 *)

Crossrefs

Sequence in context: A224234 A305368 A197412 * A113794 A086774 A153461

Adjacent sequences: A275370 A275371 A275372 * A275374 A275375 A275376

Keyword

nonn,cons

Author

Jean-François Alcover, Jul 25 2016

Status

approved