A019881 -id:A019881 - cp's OEIS frontend

A179296 Decimal expansion of circumradius of a regular dodecahedron with edge length 1.

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 09 2010

Dodecahedron: A three-dimensional figure with 12 faces, 20 vertices, and 30 edges.

Appears as a coordinate in a degree-7 quadrature formula on 12 points over the unit circle [Stroud & Secrest]. - R. J. Mathar, Oct 12 2011

			1.40125853844407354467667793532206799444393173977549286366084518639135...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Chai Wah Wu, Table of n, a(n) for n = 1..10001
A. H. Stroud and Don Secrest, Approximate integration formulas for certain spherically symmetric regions, Math. Comp. 17 (82) (1963) 105.
Wikipedia, Dodecahedron.
Wolfram Alpha, Dodecahedron.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A102208, A102769, A131595, A179290, A179292, A179294.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A019881 (icosahedron), A187110 (tetrahedron). - Stanislav Sykora, Feb 10 2014

RealDigits[(Sqrt[3]+Sqrt[15])/4, 10, 175][[1]]

(1+sqrt(5))*sqrt(3)/4 \\ Stefano Spezia, Jan 27 2025

Equals (sqrt(3) + sqrt(15))/4 = sqrt((9 + 3*sqrt(5))/8).
The minimal polynomial is 16*x^4 - 36*x^2 + 9. - Joerg Arndt, Feb 05 2014
Equals (sqrt(3)/2) * phi = A010527 * A001622. - Amiram Eldar, Jun 02 2023

A179591 Decimal expansion of the surface area of pentagonal cupola with edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Pentagonal cupola: 15 vertices, 25 edges, and 12 faces.

			16.5797497529881970460940463443632246181026360961176551818747440...

Wolfram Alpha, Johnson solid 5
Index entries for algebraic numbers, degree 8.

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587, A179588, A179589, A179590.

RealDigits[N[(20+Sqrt[10*(80+31*Sqrt[5]+Sqrt[2175+930*Sqrt[5]])])/4,200]]

Digits of (20+sqrt(10*(80+31*sqrt(5)+sqrt(2175+930*sqrt(5)))))/4.

A188593 Decimal expansion of (diagonal)/(shortest side) of a golden rectangle.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 04 2011

A rectangle of length L and width W is a golden rectangle if L/W = r = (1+sqrt(5))/2. The diagonal has length D = sqrt(L^2+W^2), so D/W = sqrt(r^2+1) = sqrt(r+2).
Largest root of x^4 - 5x^2 + 5. - Charles R Greathouse IV, May 07 2011
This is the case n=10 of (Gamma(1/n)/Gamma(2/n))*(Gamma((n-1)/n)/Gamma((n-2)/n)) = 2*cos(Pi/n). - Bruno Berselli, Dec 13 2012
Edge length of a pentagram (regular star pentagon) with unit circumradius. - Stanislav Sykora, May 07 2014
The ratio diagonal/side of the shortest diagonal in a regular 10-gon. - Mohammed Yaseen, Nov 04 2020

			1.902113032590307144232878666758764286811397268251...

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Michael Penn, On the fifth root of the identity matrix, YouTube video, 2022.
Eric Weisstein's World of Mathematics, Golden Rectangle.
Eric Weisstein's World of Mathematics, Pentagram.
Index entries for algebraic numbers, degree 4.

Cf. A001622 (decimal expansion of the golden ratio), A019881.
Cf. A188594 (D/W for the silver rectangle, r=1+sqrt(2)).
Cf. A090771, A195693, A195723, A296184.

SetDefaultRealField(RealField(100)); Sqrt((5+Sqrt(5))/2); // G. C. Greubel, Nov 02 2018

r = (1 + 5^(1/2))/2; RealDigits[(2 + r)^(1/2), 10, 130][[1]]
RealDigits[Sqrt[GoldenRatio+2],10,130][[1]] (* Harvey P. Dale, Oct 27 2023 *)

PARI
```
sqrt((5+sqrt(5))/2)
```

Equals 2*A019881. - Mohammed Yaseen, Nov 04 2020
Equals csc(A195693) = sec(A195723). - Amiram Eldar, May 28 2021
Equals i^(1/5) + i^(-1/5). - Gary W. Adamson, Jul 08 2022
Equals sqrt(2 + phi) = sqrt(A296184), with phi = A001622. - Wolfdieter Lang, Aug 28 2022
Equals Product_{k>=0} ((10*k + 2)*(10*k + 8))/((10*k + 1)*(10*k + 9)). - Antonio Graciá Llorente, Feb 24 2024
Equals Product_{k>=1} (1 - (-1)^k/A090771(k)). - Amiram Eldar, Nov 23 2024

A019890 Decimal expansion of sine of 81 degrees.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Also the real part of i^(1/10). - Stanislav Sykora, Apr 25 2012
Equals sin(9*Pi/20). - Wesley Ivan Hurt, Sep 01 2014
An algebraic number of degree 8 and denominator 2. - Charles R Greathouse IV, Aug 27 2017

			0.98768834059513772619004024769343726075840686158988043492390480163...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Trigonometric constants expressed in real radicals
Index entries for algebraic numbers, degree 8

Digits:=100: evalf(sin(9*Pi/20)); # Wesley Ivan Hurt, Sep 01 2014

RealDigits[Sin[81 Degree],10,120][[1]] (* Harvey P. Dale, Aug 11 2012 *)

sin(9*Pi/20) \\ Charles R Greathouse IV, Aug 27 2017

Equals cos(Pi/20) = sqrt((1+A019881)/2) = sqrt(1-A019818^2) = sqrt(5-sqrt(5))*(sqrt(5)+sqrt(5+2*sqrt(5)))/(4*sqrt(5)). - R. J. Mathar, Jun 18 2006
Equals A019863 * A019872 + A019836 * A019845 = A019887 * A019896 + A019812 * A019821. - R. J. Mathar, Jan 27 2021
Root of 256*x^8 -512*x^6 +304*x^4 -48*x^2+1=0. - R. J. Mathar, Aug 29 2025
Equals 2F1(-1/10,1/10;1/2;1/2). - R. J. Mathar, Aug 31 2025

A179588 Decimal expansion of the surface area of square cupola with edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Square cupola: 12 vertices, 20 edges, and 10 faces.

			11.56047793231506739113082378992526852408214900456427677440...

Wolfram Alpha, Johnson solid 4
Index entries for algebraic numbers, degree 4

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587.

Mathematica
```
RealDigits[N[7+2*Sqrt[2]+Sqrt[3],200]]
```

Digits of 7 + 2*sqrt(2) + sqrt(3).

A187110 Decimal expansion of sqrt(3/8).

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 05 2011

Apart from leading digits, the same as A174925.

Radius of the circumscribed sphere (congruent with vertices) for a regular tetrahedron with unit edges. - Stanislav Sykora, Nov 20 2013

			sqrt(3/8)=0.61237243569579452454932101867647284799148687016417..

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Michael Penn, Maximize the area of one ellipse!, YouTube video, 2020.
Wikipedia, Tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. A002194, A010464, A010466, A020781, A040001, A040005, A115754, A301755.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A019881 (icosahedron), A179296 (dodecahedron). - Stanislav Sykora, Feb 10 2014

Mathematica
```
RealDigits[N[Sqrt[3/8],300]][[1]]
```

sqrt(3/8) \\ Charles R Greathouse IV, Mar 16 2022

Equals A010464/4. - Stefano Spezia, Jan 26 2025

Equals 3*A020781 = A115754/2 = sqrt(A301755). - Hugo Pfoertner, Jan 26 2025

A339259 Decimal expansion of the volume of the regular icosahedron inscribed in the unit sphere.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Nov 29 2020

			2.536150710120409525643838222345019049081863024335333926526148385147...

Index entries for algebraic numbers, degree 4.

Cf. A001622, A019881, A102208.

Cf. A118273 (cube), A122553 (regular octahedron), A363437 (regular tetrahedron), A363438 (regular dodecahedron).

RealDigits[4 * Sqrt[GoldenRatio + 2]/3, 10, 120][[1]] (* Amiram Eldar, Jun 02 2023 *)

PARI
```
4/3*sqrt(2+(1+sqrt(5))/2)
```

Equals 4*sqrt(2 + phi)/3 where phi = A001622.

Equals A102208 / A019881 ^ 3. - Amiram Eldar, Jun 02 2023

A019916 Decimal expansion of tan(Pi/10) (angle of 18 degrees).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

In a regular pentagon inscribed in a unit circle this is the cube of the length of the side divided by 5: (1/5)*(sqrt(3 - phi))^3 with phi from A001622. - Wolfdieter Lang, Jan 08 2018
Quartic number of denominator 5 and minimal polynomial 5x^4 - 10x^2 + 1. - Charles R Greathouse IV, May 13 2019
The other positive root of the minimal polynomial is A019952. - R. J. Mathar, Sep 06 2025

			0.3249196962329063261558714122151344649549034715214751003078047191...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Exact trigonometric constants
Index entries for algebraic numbers, degree 4.

Cf. A001622, A019827 (sin(Pi/10)), A019881 (cos(Pi/10)).

RealDigits[Tan[18 Degree],10,120][[1]] (* Harvey P. Dale, Mar 07 2012 *)

tan(Pi/10) \\ Michel Marcus, Jan 08 2018

polrootsreal(5*x^4-10*x^2+1)[3] \\ Charles R Greathouse IV, Feb 04 2025

Equals A019827/A019881 = 1/A019970 = 1/sqrt(5+2*sqrt(5)). - R. J. Mathar, Jul 26 2010
Equals tan((phi - 1)/sqrt(2 + phi)) = (1/5)*(sqrt(3 - phi))^3 = (3 - phi)*sqrt(3 - phi)/5 = sqrt(7 - 4*phi)/(2*phi - 1), with phi from A001622. - Wolfdieter Lang, Jan 08 2018
Equals Product_{k>=0} ((5*k + 1)/(5*k + 4))^(-1)^(k) = Product_{k>=0} A090771(k)/A090773(k). - Antonio Graciá Llorente, Mar 24 2024
Equals A019845/(1+A019863). - R. J. Mathar, Sep 06 2025

A179589 Decimal expansion of the circumradius of square cupola with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Square cupola: 12 vertices, 20 edges, and 10 faces.

			1.398966325965906702031540539431998764673522563866238879930...

Wolfram Alpha, Johnson solid 4
Index entries for algebraic numbers, degree 4

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587, A179588.

Mathematica
```
RealDigits[N[Sqrt[5+2*Sqrt[2]]/2,200]]
```

Digits of sqrt(5+2*sqrt(2))/2.

A232735 Decimal expansion of the real part of I^(1/7), or cos(Pi/14).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding imaginary part is in A232736.

Root of the equation -7 + 56*x^2 - 112*x^4 + 64*x^6 = 0. - Vaclav Kotesovec, Apr 04 2021

			0.974927912181823607018131682993931217232785800619997437648...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
A. Arman, A. Bondarenko, and A. Prymak, Convex bodies of constant width with exponential illumination number, arXiv:2304.10418 [math.MG], 2023.
Index entries for algebraic numbers, degree 6

Cf. A232736 (imaginary part), A010503 (real(I^(1/2))), A010527 (real(I^(1/3))), A144981 (real(I^(1/4))), A019881 (real(I^(1/5))), A019884 (real(I^(1/6))), A232737 (real(I^(1/8))), A019889 (real(I^(1/9))), A019890 (real(I^(1/10))).

R:= RealField(100); Cos(Pi(R)/14); // G. C. Greubel, Sep 19 2022

RealDigits[Cos[Pi/14],10,120][[1]] (* Harvey P. Dale, Dec 15 2018 *)

numerical_approx(cos(pi/14), digits=120) # G. C. Greubel, Sep 19 2022

2*this^2 -1 = A073052. - R. J. Mathar, Aug 29 2025

Equals 2F1(-1/14,1/14;1/2;1) . - R. J. Mathar, Aug 31 2025

cp's OEIS Frontend

A179296 Decimal expansion of circumradius of a regular dodecahedron with edge length 1.

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A179591 Decimal expansion of the surface area of pentagonal cupola with edge length 1.

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A188593 Decimal expansion of (diagonal)/(shortest side) of a golden rectangle.

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Magma

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A019890 Decimal expansion of sine of 81 degrees.

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A179588 Decimal expansion of the surface area of square cupola with edge length 1.

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A187110 Decimal expansion of sqrt(3/8).

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A339259 Decimal expansion of the volume of the regular icosahedron inscribed in the unit sphere.

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