A019819 -id:A019819 - cp's OEIS frontend

A280633 Decimal expansion of 18*sin(Pi/18).

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

Offset: 1

Rick L. Shepherd, Jan 06 2017

The ratio of the perimeter of a regular 9-gon (nonagon) to its diameter (largest diagonal).

Also least positive root of x^3 - 243x + 729.

			3.125667198004746279330899281847666328006762189313248970252344806377184798...

Index entries for algebraic numbers, degree 3.

Cf. For other n-gons: A010466 (n=4), 10*A019827 (n=5, 10), A280533 (n=7),A280585 (n=8), A280725(n=11), A280819 (n=12).

evalf(18*sin(Pi/18),100); # Wesley Ivan Hurt, Feb 01 2017

RealDigits[18*Sin[Pi/18],10,120][[1]] (* Harvey P. Dale, Dec 02 2018 *)

PARI
```
18*sin(Pi/18)
```

18*A019819.

A019978 Decimal expansion of tangent of 80 degrees.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			tan(4*Pi/9) = 5.67128183...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Index entries for algebraic numbers, degree 6

RealDigits[Tan[80 Degree],10,120][[1]] (* Harvey P. Dale, Jan 04 2013 *)

Largest positive of the 6 real-valued roots of x^6-33*x^4+27*x^2-3=0. - R. J. Mathar, Aug 31 2025

Equals A019889/A019819. - R. J. Mathar, Aug 31 2025

A133749 Decimal expansion of -2cos((2Pi)/9) + 2sqrt(3)sin((2*Pi)/9).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Sep 23 2007

The distance between the centers of 2 unit-radius spheres such that the volume of the overlap is equal to half the volume of each sphere. - Amiram Eldar, May 31 2021

			0.69459271066772139540...

Eric Weisstein's World of Mathematics, Sphere-Sphere Intersection.

Cf. A019819.

RealDigits[-2*Cos[(2*Pi)/9] + 2*Sqrt[3]*Sin[(2*Pi)/9], 10, 100][[1]] (* Vaclav Kotesovec, Aug 16 2015 *)

4*sin(Pi/18) \\ Gleb Koloskov, Feb 28 2021

Equals 4*sin(Pi/18) = 4*A019819. - Gleb Koloskov, Feb 28 2021

A343055 Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Apr 04 2021

An algebraic number of degree 16 and denominator 2. - Charles R Greathouse IV, Jan 09 2022

			0.09801714032956060199419...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Penn, The exact value of sin 5⅝°??, YouTube video, 2021.
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 16

sin(Pi/m): A010527 (m=3), A010503 (m=4), A019845 (m=5), A323601 (m=7), A182168 (m=8), A019829 (m=9), A019827 (m=10), A019824 (m=12), A232736 (m=14), A019821 (m=15), A232738 (m=16), A241243 (m=17), A019819 (m=18), A019818 (m=20), A343054 (m=24), A019815 (m=30), this sequence (m=32), A019814 (m=36).

RealDigits[Sin[Pi/32], 10, 100, -1][[1]] (* Amiram Eldar, Apr 27 2021 *)

PARI
```
imag(I^(1/16))
```
PARI
```
sin(Pi/32)
```
PARI
```
sqrt(2-sqrt(2+sqrt(2+sqrt(2))))/2
```

numerical_approx(sin(pi/32), digits=123) # G. C. Greubel, Sep 30 2022

Equals (1/2) * sqrt(2-sqrt(2+sqrt(2+sqrt(2)))).
One of the 16 real roots of -128*x^2 +2688*x^4 -21504*x^6 +84480*x^8 +32768*x^16 -131072*x^14 +212992*x^12 -180224*x^10 +1 =0. - R. J. Mathar, Aug 29 2025
Equals A232738/(2*A343056). - R. J. Mathar, Sep 05 2025

A359837 Decimal expansion of the unsigned ratio of similitude between an equilateral reference triangle and its first Morley triangle.

Original entry on oeis.org

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

Offset: 0

Frank M Jackson, Jan 14 2023

The first Morley triangle of any reference triangle is always equilateral. Therefore a reference equilateral triangle and its first Morley triangle will be in a homothetic relationship. This sequence is the real terms of a constant that is the ratio of similitude of the homothety. The constant is negative.

If an equilateral triangle has a side a, a circumradius R and a first Morley triangle with side a', then a = R*sqrt(3) and a' = 8*R*(sin(Pi/9))^3, so the ratio of similitude between the two triangles is a'/a = (8/sqrt(3)) * (sin(Pi/9))^3. - Bernard Schott, Feb 06 2023

			0.1847925309040953727013520475722037558709135597926517172356...

Wikipedia, Morley's trisector theorem.
Index entries for algebraic numbers, degree 3.

Cf. A019819, A019829, A019879, A020760.

RealDigits[Sin[Pi/18]/Cos[Pi/9], 10, 100][[1]]
N[Solve[x^3 + 3*x^2 - 6*x + 1 == 0, {x}][[2]], 90]

sin(Pi/18)/cos(Pi/9) \\ Michel Marcus, Jan 15 2023

Equals sin(Pi/18)/cos(Pi/9).
A root of x^3+3*x^2-6*x+1.
Equals A019819/A019879. - Michel Marcus, Jan 15 2023
Equals 8 * A020760 * A019829^3. - Bernard Schott, Feb 06 2023

A383859 Central angle of the solution of the Tammes problem for 7 points on the sphere (in radians).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, May 12 2025

			1.3590798976326601418850028816473327537..

Laslo Hars, Numerical solutions of the Tammes problem for up to 60 points, Nov 2020, N=7.
K. Schutte and B. L. van der Waerden, Auf welcher Kugel haben 5, 6, 7, 8 oder 9 Punkte mit Mindestabstand eins Platz?, Math. Ann. 123 (1951) 96-124.
Wikipedia, Tammes problem

Cf. A019819, A019669 (N=6), A381756 (N=8), A137914 (N=9), A340918 (N=10), A105199 (N=11 and N=12). A217695 (N=13), A383860 (N=14), A383861 (N=24).

cos(4*Pi/9) ; %/(1-%) ; arccos(%) ; evalf(%,120) ;

cos( this ) = cos phi/(1- cos phi) where cos(phi)=A019819.

cp's OEIS Frontend

A280633 Decimal expansion of 18*sin(Pi/18).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A019978 Decimal expansion of tangent of 80 degrees.

Views

Author

Keywords

Examples

Links

Programs

Mathematica

Formula

A133749 Decimal expansion of -2*cos((2*Pi)/9) + 2*sqrt(3)*sin((2*Pi)/9).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A343055 Decimal expansion of the imaginary part of i^(1/16), or sin(Pi/32).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Sage

Formula

A359837 Decimal expansion of the unsigned ratio of similitude between an equilateral reference triangle and its first Morley triangle.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A383859 Central angle of the solution of the Tammes problem for 7 points on the sphere (in radians).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A133749 Decimal expansion of -2cos((2Pi)/9) + 2sqrt(3)sin((2*Pi)/9).