cp's OEIS Frontend

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.

Showing 1-10 of 13 results. Next

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

Views

Author

Clark Kimberling, Jul 11 2020

Keywords

Comments

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

Examples

			arclength = 2.3098814600100572608866337793136248461119964...

Links

Crossrefs

Cf. A336073, A003957, A019669.
Cf. A086118, A361605.

Programs

  • Mathematica
    k = 3; s = s /. FindRoot[(2 Pi - s + Sin[s])/(s - Sin[s]) == k, {s, 2}, WorkingPrecision -> 200]
RealDigits[s][[1]]
  • PARI
    d=solve(x=0,1,cos(x)-x); d+Pi/2 \\ Gleb Koloskov, Feb 21 2021

Formula

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

A336082 Decimal expansion of the arclength on the unit circle such that the corresponding chord separates the interior into segments having 2 = ratio of segment areas; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 11 2020

Keywords

Comments

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.

Examples

			arclength = 2.605325674600902685700194315412975801441...

Crossrefs

Cf. A336073.

Programs

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

A336074 Decimal expansion of the ratio of segment areas for arclength Pi/6 on the unit circle; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 10 2020

Keywords

Comments

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.

Examples

			ratio = 265.25047901351764771254818549254008953843516090523...

Crossrefs

Cf. A336073.

Programs

  • Mathematica
    s = Pi/6; r = N[(2 Pi - s + Sin[s])/(s - Sin[s]), 200]
RealDigits[r][[1]]

Formula

ratio = (2*Pi - s + sin(s))/(s - sin(s)), where s = Pi/6.

A336075 Decimal expansion of the ratio of segment areas for arclength Pi/5 on the unit circle; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 11 2020

Keywords

Comments

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.

Examples

			ratio = 154.01300539334118250202674320597591102537...

Crossrefs

Cf. A336073.

Programs

  • Mathematica
    s = Pi/5; r = N[(2 Pi - s + Sin[s])/(s - Sin[s]), 200]
RealDigits[r][[1]]

Formula

ratio = (2*Pi - s + sin(s))/(s - sin(s)), where s = Pi/5.

A336076 Decimal expansion of (7*Pi + 2*sqrt(2)) / (Pi - 2*sqrt(2)).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 11 2020

Keywords

Comments

Decimal expansion of the ratio of segment areas for arclength Pi/4 on the unit circle. In general, 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.

Examples

			79.2538559129532960392332048980091137833110385495...

Links

Crossrefs

Cf. A336073.

Programs

  • Maple
    s := Pi/4 ; sss := s-sin(s) ; evalf( 2*Pi/sss -1 ) ; # R. J. Mathar, Sep 02 2020
  • Mathematica
    s = Pi/4; r = N[(2 Pi - s + Sin[s])/(s - Sin[s]), 200]
RealDigits[r][[1]]

Formula

Equals (2*Pi - s + sin(s))/(s - sin(s)), where s = Pi/4 = A003881.

A336077 Decimal expansion of (10*Pi + 3*sqrt(3)) / (2*Pi - 3*sqrt(3)).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 11 2020

Keywords

Comments

Decimal expansion of the ratio of segment areas for arclength Pi/3 on the unit circle. In general, 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.

Examples

			33.68074644435052842991251795285920080736045858...

Links

Crossrefs

Cf. A336073.

Programs

  • Maple
    s := Pi/3 ;
sss := s-sin(s) ;
evalf( 2*Pi/sss -1 ) ; # R. J. Mathar, Sep 02 2020
  • Mathematica
    s = Pi/3; r = N[(2 Pi - s + Sin[s])/(s - Sin[s]), 200]
RealDigits[r][[1]]

Formula

Equals (2*Pi - s + sin(s))/(s - sin(s)), where s = Pi/3 = A019670.

A336078 Decimal expansion of (3*Pi + 2)/(Pi - 2).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 11 2020

Keywords

Comments

Decimal expansion of the ratio of segment areas for arclength Pi/2 on the unit circle. In general, 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.

Examples

			10.00775357553643464481563880604581551700159202722296...

Links

Crossrefs

Cf. A019669, A336073.

Programs

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

Formula

Equals (2*Pi - s + sin(s))/(s - sin(s)), where s = Pi/2 = A019669.

A336079 Decimal expansion of the ratio of segment areas for arclength 1 on the unit circle; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 11 2020

Keywords

Comments

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.

Examples

			ratio = 38.63429218030340056508641778759493689126124881320...

Crossrefs

Cf. A336073.

Programs

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

Formula

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

A336080 Decimal expansion of the ratio of segment areas for arclength 2 on the unit circle; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 11 2020

Keywords

Comments

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.

Examples

			ratio = 4.760677073396245684037489831539316359481234621068492...

Crossrefs

Cf. A336073.

Programs

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

Formula

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

A336081 Decimal expansion of the ratio of segment areas for arclength 3 on the unit circle; see Comments.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jul 11 2020

Keywords

Comments

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.

Examples

			ratio = 1.19777861431518297091106473299080089125851089...

Crossrefs

Cf. A336073.

Programs

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

Formula

ratio = (2*Pi - s + sin(s))/(s - sin(s)), where s = 3.
Showing 1-10 of 13 results. Next

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