A086118 -id:A086118 - cp's OEIS frontend

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.

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

A128463 Decimal expansion of area of Gerver's sofa.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, May 05 2007

			2.219531668871967...

Steven R. Finch, Moving Sofa Constant, Sect. 8.12 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 519-523, 2003.

Jineon Baek, Optimality of Gerver's Sofa, arXiv:2411.19826 [math.MG], 2024.
J. L. Gerver, On Moving a Sofa Around a Corner, Geometriae Dedicata 42, 267-283, 1992.
Yoav Kallus and Dan Romik, Improved upper bounds in the moving sofa problem, arXiv preprint arXiv:1706.06630 [math.MG], 2017.
Kuangdai Leng, Jia Bi, Jaehoon Cha, Samuel Pinilla, and Jeyan Thiyagalingam, Deep Learning Evidence for Global Optimality of Gerver's Sofa, arXiv:2407.11106 [cs.LG], 2024. See p. 16.
Dan Romik, Differential equations and exact solutions in the moving sofa problem, arXiv:1606.08111 [math.DG], 2016.
Dan Romik, Differential equations and exact solutions in the moving sofa problem, Experimental Mathematics (2017): 1-15.
Eric Weisstein's World of Mathematics, Gerver Sofa.
Eric Weisstein's World of Mathematics, Moving Sofa Constant.
Eric Weisstein's World of Mathematics, Moving Sofa Problem.
Wikipedia, Moving sofa problem

Cf. A086118.

More terms from Jinyuan Wang, Dec 04 2024

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).

A330934 Decimal expansion of the area of a sofa that can be moved around a 90-degree turn both to the right and to the left in a hallway of unit width.

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jan 03 2020

According to Dan Romik, this may be the largest possible area of such a sofa. He gives the closed-form formula below for the area of this shape which consists of "18 distinct pieces, each of which is given by a separate formula obtained as the solution of some differential equation." See the D. Romik link for a picture of this shape and animations of this and related sofas.

			1.644955218425440851668809347600633685194252864098962636889345708010329...

Dan Romik, The Moving Sofa Problem.
Dan Romik, Differential equations and exact solutions in the moving sofa problem, Experimental Mathematics, Vol. 27, No. 3 (2018), pp. 316-330; arXiv preprint, arXiv:1606.08111 [math.DG], 2016.
Eric Weisstein's World of Mathematics, Moving Sofa Problem.
Wikipedia, Moving sofa problem.

Cf. A086118, A128463.

RealDigits[(3 + 2*Sqrt[2])^(1/3) + (3 - 2*Sqrt[2])^(1/3) - 1 + ArcTan[((Sqrt[2] + 1)^(1/3) - (Sqrt[2] - 1)^(1/3))/2], 10, 120][[1]] (* Amiram Eldar, Jun 18 2023 *)

{default(realprecision, 200);
my(sr2 = sqrt(2)); (3+2*sr2)^(1/3) + (3-2*sr2)^(1/3) - 1 + atan(((sr2+1)^(1/3) - (sr2-1)^(1/3))/2)}

Equals (3 + 2*sqrt(2))^(1/3) + (3 - 2*sqrt(2))^(1/3) - 1 + atan(((sqrt(2) + 1)^(1/3) - (sqrt(2) - 1)^(1/3))/2) [D. Romik].

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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.

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A128463 Decimal expansion of area of Gerver's sofa.

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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).

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A330934 Decimal expansion of the area of a sofa that can be moved around a 90-degree turn both to the right and to the left in a hallway of unit width.

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Mathematica

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Formula