This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A361605 #12 Mar 17 2023 08:08:43
%S A361605 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,
%T A361605 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,
%U A361605 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
%N 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).
%C A361605 The corresponding value in degrees is 37.0071439021...
%H A361605 Math Stackexchange, <a href="https://math.stackexchange.com/questions/464419/mean-value-of-the-rotation-angle-is-126-5%C2%B0">Mean value of the rotation angle is 126.5 degrees</a>, 2013.
%H A361605 Marc B. Reynolds, <a href="https://marc-b-reynolds.github.io/quaternions/2017/11/10/AveRandomRot.html">Volume element of SO(3) and average uniform random rotation angle</a>, 2017.
%H A361605 Hansklaus Rummler, <a href="https://doi.org/10.1007/BF03025318">On the distribution of rotation angles how great is the mean rotation angle of a random rotation?</a>, The Mathematical Intelligencer, Vol. 24, No. 4 (2002), pp. 6-11; <a href="https://core.ac.uk/download/pdf/159153606.pdf">alternative link</a>.
%H A361605 Eugene Salamin, <a href="https://theworld.com/~sweetser/quaternions/ps/stanfordaiwp79-salamin.pdf">Application of quaternions to computations with rotations</a>, Working Paper, Stanford AI Lab, 1979.
%F A361605 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.
%F A361605 Equals sqrt((Pi^4 - 48)/3)/(2*Pi).
%e A361605 0.64589650785149948235874138426552716216750326306111...
%t A361605 RealDigits[Sqrt[(Pi^4 - 48)/3]/(2*Pi), 10, 100][[1]]
%o A361605 (PARI) sqrt((Pi^4 - 48)/3)/(2*Pi)
%Y A361605 Cf. A086118 (mean), A336083 (median).
%K A361605 nonn,cons
%O A361605 0,1
%A A361605 _Amiram Eldar_, Mar 17 2023