A361605 -id:A361605 - cp's OEIS frontend

A086118 Decimal expansion of Pi/2 + 2/Pi.

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

Offset: 1

Eric W. Weisstein, Jul 10 2003

Lower limit in the moving sofa problem.

This is the expected angle of a random rotation [Salamin]. - Bill Gosper, Oct 20 2013

			2.2074160991624779623068567451298088902364232826493787...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, section 8.12, pp. 519-523.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Steven R. Finch, Moving Sofa Constant, Mathcad Library. [Wayback Machine link]
Math Stackexchange, Mean value of the rotation angle is 126.5 degrees, 2013.
Eugene Salamin, Application of quaternions to computation with rotations, Working Paper, Stanford AI Lab, 1979.
Marc B. Reynolds, Volume element of SO(3) and average uniform random rotation angle, 2017.
Hansklaus Rummler, On the distribution of rotation angles how great is the mean rotation angle of a random rotation?, The Mathematical Intelligencer, Vol. 24, No. 4 (2002), pp. 6-11; alternative link.
Eric Weisstein's World of Mathematics, Hammersley Sofa.
Eric Weisstein's World of Mathematics, Moving Sofa Problem.
Wikipedia, Moving sofa problem.
Index entries for transcendental numbers.

Cf. A128463 (a better lower bound), A336083, A361605.

RealDigits[Pi/2 + 2/Pi, 10, 100][[1]] (* Amiram Eldar, Mar 17 2023 *)

2/Pi+Pi/2 \\ Charles R Greathouse IV, Oct 01 2022

A336083 Decimal expansion of the arclength on the unit circle such that the corresponding chord separates the interior into segments having 3 = ratio of segment areas; see Comments.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Jul 11 2020

Suppose that s in (0,Pi) is the length of an arc of the unit circle. The associated chord separates the interior into two segments. Let A1 be the area of the larger and A2 the area of the smaller. The term "ratio of segment areas" means A1/A2. See A336073 for a guide to related sequences.

Equals the median of the probability distribution function of angles of random rotations in 3D space uniformly distributed with respect to the Haar measure, i.e., the solution x to Integral_{t=0..x} ((1 - cos(t))/Pi) dt = 1/2 (see Reynolds, 2017; cf. A086118, A361605). - Amiram Eldar, Mar 17 2023

			arclength = 2.3098814600100572608866337793136248461119964...

Marc B. Reynolds, Volume element of SO(3) and average uniform random rotation angle, 2017.

Cf. A336073, A003957, A019669.

Cf. A086118, A361605.

k = 3; s = s /. FindRoot[(2 Pi - s + Sin[s])/(s - Sin[s]) == k, {s, 2}, WorkingPrecision -> 200]
RealDigits[s][[1]]

d=solve(x=0,1,cos(x)-x); d+Pi/2 \\ Gleb Koloskov, Feb 21 2021

Equals d+Pi/2 = A003957 + A019669, where d is the Dottie number. - Gleb Koloskov, Feb 21 2021

cp's OEIS Frontend

A086118 Decimal expansion of Pi/2 + 2/Pi.

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