A361618 Decimal expansion of the mean of the distribution of the least of the nine acute angles between pairs of edges of two randomly disoriented cubes (in radians).

4, 0, 4, 2, 8, 3, 3, 4, 5, 0, 4, 4, 8, 9, 3, 5, 8, 5

Offset

0,1

Comments

The angle in degrees is 23.1637293985...

Links

Table of n, a(n) for n=0..17.

Amiram Eldar, Mathematica code for A361618 and A361619.

J. K. Mackenzie, Second Paper on Statistics Associated with the Random Disorientation of Cubes, Biometrika, Vol. 45, No. 1-2 (1958), pp. 229-240.

J. K. Mackenzie and M. J. Thomson, Some Statistics Associated with the Random Disorientation of Cubes, Biometrika, Vol. 44, No. 1-2 (1957), pp. 205-210.

Wikipedia, Misorientation.

Formula

Equals Integral_{t=0..arccos(2/3)} t * P(t) dt, where P(t) = P1(t) if 0 <= t <= Pi/4, and P1(t) + P2(t) if Pi/4 < t <= arccos(2/3), and where

P1(t) = (288/Pi^2) * sin(t) * (Pi^2/32 - Integral_{x=0..Pi/4} arcsin(tan(t/2)*cos(x)) dx),

P2(t) = -(288/Pi^2) * sin(t) * (Pi*g(t)/4 - Integral_{x=Pi/4-g(t)..Pi/8+g(t)} arcsin(tan(t/2)*cos(x)) dx),

g(t) = arcsin(sqrt(((sqrt(2) + 1)^2 * tan(t/2)^2 - 1)/(4 * sqrt(2) * tan(t/2)^2))).

Example

0.404283345044893585...

Mathematica

See the program in the links section.

Crossrefs

Cf. A228496, A361601, A361619, A361620, A361621.

Sequence in context: A081087 A135031 A238002 * A367873 A016680 A062524

Adjacent sequences: A361615 A361616 A361617 * A361619 A361620 A361621

Keyword

nonn,cons,more

Author

Amiram Eldar, Mar 18 2023

Status

approved