A336083 -id:A336083 - cp's OEIS frontend

A336073 Decimal expansion of the ratio of segment areas for arclength 1/3 on the unit circle; see Comments.

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

Offset: 4

Clark Kimberling, Jul 10 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.
*****************
Guide to related sequences:
arclength,s   ratio, A1/A2
1/3           A336073
Pi/6          A336074
Pi/5          A336075
Pi/4          A336076
Pi/3          A336077
Pi/2          A336078
1             A336079
2             A336080
3             A336081
*****************
ratio, A1/A2  arclength, s
2             A336082
3             A336083
4             A336084
5             A336085
1/2           A336086

			ratio = 1022.54737393604920361975925805839994393435790826122033281

Cf. A336059-A336086.

s = 1/3; r = N[(2 Pi - s + Sin[s])/(s - Sin[s]), 200]
RealDigits[r][[1]]

2*Pi/(1/3 - sin(1/3)) - 1 \\ Charles R Greathouse IV, Feb 22 2025

ratio = (2*Pi - s + sin(s))/(s - sin(s)), where s = 1/3.

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

Original entry on oeis.org

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

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).

Original entry on oeis.org

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

Amiram Eldar, Mar 17 2023

The corresponding value in degrees is 37.0071439021...

			0.64589650785149948235874138426552716216750326306111...

Math Stackexchange, Mean value of the rotation angle is 126.5 degrees, 2013.
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.
Eugene Salamin, Application of quaternions to computations with rotations, Working Paper, Stanford AI Lab, 1979.

Cf. A086118 (mean), A336083 (median).

RealDigits[Sqrt[(Pi^4 - 48)/3]/(2*Pi), 10, 100][[1]]

PARI
```
sqrt((Pi^4 - 48)/3)/(2*Pi)
```

Equals sqrt( - ^2), where = 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).

cp's OEIS Frontend

A336073 Decimal expansion of the ratio of segment areas for arclength 1/3 on the unit circle; see Comments.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula