A395734 Decimal expansion of the mean width of an oloid formed by two unit circles.

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

Offset

1,1

Comments

The mean curvature of the oloid is 2*Pi times this constant: 13.7644293270030696543... .

Links

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

Uwe Bäsel, The Mean Width and Integral Geometric Properties of the Oloid, Journal for Geometry and Graphics, Vol. 22, No. 2 (2018), pp. 149-161. See Theorem 1, p. 154.

Robert Ferréol, Oloid, MathCurve.

Eric Weisstein's World of Mathematics, Complete Elliptic Integral of the First Kind.

Eric Weisstein's World of Mathematics, Oloid.

Wikipedia, Oloid.

Formula

Equals (3 * K(sqrt(3)/2) + 3*Pi^2/2 - 4 * Integral_{x=0..Pi/2} arccos(cos(x)/(1+cos(x))) dx) / (2*Pi), where K(x) is the complete elliptic integral of the first kind.

Example

2.190676966231588766332630494363152369733991811525247...

Mathematica

RealDigits[(3 * EllipticK[3/4] + 3*Pi^2/2 - 4 * NIntegrate[ArcCos[Cos[x]/(1 + Cos[x])], {x, 0, Pi/2}, WorkingPrecision -> 120]) / (2*Pi)][[1]]

Prog

(PARI) (3 * ellK(sqrt(3)/2) + 3*Pi^2/2 - 4 * intnum(x = 0, Pi/2, acos(cos(x)/(1+cos(x))))) / (2*Pi)

Crossrefs

Cf. A215447 (volume), A249283.

Cf. A289502 (regular tetrahedron), A152623 (cube), A395731 (regular octahedron), A395732 (regular dodecahedron), A395733 (regular icosahedron).

Sequence in context: A230582 A011186 A388364 * A078088 A206243 A371930

Adjacent sequences: A395731 A395732 A395733 * A395735 A395736 A395737

Keyword

nonn,cons

Author

Amiram Eldar, May 05 2026

Status

approved