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A361605
Decimal expansion of the standard deviation of the probability distribution function of angles of random rotations in 3D space uniformly distributed with respect to Haar measure (in radians).
2
6, 4, 5, 8, 9, 6, 5, 0, 7, 8, 5, 1, 4, 9, 9, 4, 8, 2, 3, 5, 8, 7, 4, 1, 3, 8, 4, 2, 6, 5, 5, 2, 7, 1, 6, 2, 1, 6, 7, 5, 0, 3, 2, 6, 3, 0, 6, 1, 1, 1, 1, 7, 0, 2, 7, 3, 2, 9, 1, 2, 0, 4, 9, 9, 3, 8, 5, 5, 1, 4, 6, 1, 9, 3, 6, 7, 7, 7, 5, 7, 2, 1, 7, 1, 5, 2, 5, 9, 5, 1, 1, 4, 9, 1, 6, 6, 3, 5, 0, 5, 2, 1, 0, 8, 0
OFFSET
0,1
COMMENTS
The corresponding value in degrees is 37.0071439021...
FORMULA
Equals sqrt(<t^2> - <t>^2), where <t^k> = Integral_{t=0..Pi} t^k * P(t) dt, and P(t) = (1 - cos(t))/Pi is the probability distribution function of the angles in radians.
Equals sqrt((Pi^4 - 48)/3)/(2*Pi).
EXAMPLE
0.64589650785149948235874138426552716216750326306111...
MATHEMATICA
RealDigits[Sqrt[(Pi^4 - 48)/3]/(2*Pi), 10, 100][[1]]
PROG
(PARI) sqrt((Pi^4 - 48)/3)/(2*Pi)
CROSSREFS
Cf. A086118 (mean), A336083 (median).
Sequence in context: A372995 A199385 A177159 * A356479 A317866 A309977
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Mar 17 2023
STATUS
approved