A179294 -id:A179294 - cp's OEIS frontend

A019881 Decimal expansion of sin(2*Pi/5) (sine of 72 degrees).

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

Offset: 0

N. J. A. Sloane

Circumradius of pentagonal pyramid (Johnson solid 2) with edge 1. - Vladimir Joseph Stephan Orlovsky, Jul 19 2010
Circumscribed sphere radius for a regular icosahedron with unit edges. - Stanislav Sykora, Feb 10 2014
Side length of the particular golden rhombus with diagonals 1 and phi (A001622); area is phi/2 (A019863). Thus, also the ratio side/(shorter diagonal) for any golden rhombus. Interior angles of a golden rhombus are always A105199 and A137218. - Rick L. Shepherd, Apr 10 2017
An algebraic number of degree 4; minimal polynomial is 16x^4 - 20x^2 + 5, which has these smaller, other solutions (conjugates): -A019881 < -A019845 < A019845 (sine of 36 degrees). - Rick L. Shepherd, Apr 11 2017
This is length ratio of one half of any diagonal in the regular pentagon and the circumscribing radius. - Wolfdieter Lang, Jan 07 2018
Quartic number of denominator 2 and minimal polynomial 16x^4 - 20x^2 + 5. - Charles R Greathouse IV, May 13 2019
This gives the imaginary part of one of the members of a conjugate pair of roots of x^5 - 1, with real part (-1 + phi)/2 = A019827, where phi = A001622. A member of the other conjugte pair of roots is (-phi + sqrt(3 - phi)*i)/2 = (-A001622 + A182007*i)/2 = -A001622/2  + A019845*i. - Wolfdieter Lang, Aug 30 2022

			0.95105651629515357211643933337938214340569863412575022244730564443015317008...

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

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Golden Rhombus.
Wikipedia, Exact trigonometric constants.
Wikipedia, Platonic solid.
Wolfram Alpha, Johnson solid 2.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A019827, A102208, A179290 (inverse), A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A182007.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A179296 (dodecahedron), A187110 (tetrahedron). - Stanislav Sykora, Feb 10 2014

SetDefaultRealField(RealField(100)); Sqrt((5 + Sqrt(5))/8); // G. C. Greubel, Nov 02 2018

Digits:=100: evalf(sin(2*Pi/5)); # Wesley Ivan Hurt, Sep 01 2014

RealDigits[Sqrt[(5 + Sqrt[5])/8], 10, 111]  (* Robert G. Wilson v *)
RealDigits[Sin[2 Pi/5], 10, 111][[1]] (* Robert G. Wilson v, Jan 07 2018 *)

default(realprecision, 120);
real(I^(1/5)) \\ Rick L. Shepherd, Apr 10 2017

Equals sqrt((5+sqrt(5))/8) = cos(Pi/10). - Zak Seidov, Nov 18 2006
Equals 2F1(13/20,7/20;1/2;3/4) / 2. - R. J. Mathar, Oct 27 2008
Equals the real part of i^(1/5). - Stanislav Sykora, Apr 25 2012
Equals A001622*A182007/2. - Stanislav Sykora, Feb 10 2014
Equals sin(2*Pi/5) = sqrt(2 + phi)/2 = -sin(3*Pi/5), with phi = A001622  - Wolfdieter Lang, Jan 07 2018
Equals 2*A019845*A019863. - R. J. Mathar, Jan 17 2021

A179587 Decimal expansion of the volume of square cupola with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Square cupola: 12 vertices, 20 edges, and 10 faces.
Also, decimal expansion of 1 + Product_{n>0} (1-1/(4*n+2)^2). - Bruno Berselli, Apr 02 2013
Decimal expansion of 1 + (least possible ratio of the side length of one inscribed square to the side length of another inscribed square in the same non-obtuse triangle). - L. Edson Jeffery, Nov 12 2014
2*sqrt(2)/3 is the radius of the base of the maximum-volume right cone inscribed in a unit-radius sphere. - Amiram Eldar, Sep 25 2022

			1.942809041582063365867792482806465385713114583584632048784453158660...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Victor Oxman and Moshe Stupel, Why are the side lengths of the squares inscribed in a triangle so close to each other?, Forum Geometricorum, Vol. 13 (2013), 113-115.
Wolfram Alpha, Johnson solid 4
Index entries for algebraic numbers, degree 2

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A224268.

Cf. A131594 (decimal expansion of sqrt(2)/3).

RealDigits[N[1+(2*Sqrt[2])/3,200]]
(* From the second comment: *) RealDigits[N[1 + Product[1 - 1/(4 n + 2)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)

sqrt(8)/3+1 \\ Charles R Greathouse IV, Nov 14 2016

Equals (3 + 2*sqrt(2))/3.

Equals 1 + 2*A131594. - L. Edson Jeffery, Nov 12 2014

A179452 Decimal expansion of sqrt(5 + 2*sqrt(5))/2, the height of a regular pentagon and midradius of an icosidodecahedron with side length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.
Height of a regular pentagon with side length 1. - Jared Kish, Oct 16 2014
Volume of a regular decagonal prism with unit side length and height 2. - Wesley Ivan Hurt, May 04 2021

			1.53884176858762670128514528801845491200335107176889621351957812518743...

Eric Weisstein's World of Mathematics, Icosidodecahedron
Eric Weisstein's World of Mathematics, Pentagon
Index entries for algebraic numbers, degree 4

Cf. A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451.

Maple
```
sqrt(5+2*sqrt(5.))/2
```

RealDigits[Sqrt[5+2Sqrt[5]]/2,10,120][[1]] (* Harvey P. Dale, Jun 23 2017 *)

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

Equals sqrt(5+2*sqrt(5))/2.

Partially rewritten by Charles R Greathouse IV, Feb 03 2011

Edited by M. F. Hasler, Oct 16 2014

A179552 Decimal expansion of the volume of pentagonal pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Pentagonal pyramid: 6 faces, 6 vertices, and 10 edges.

Also equals the covariance-matrix eigenvalue of the regular icosahedron with unit edge lengths. - Chittaranjan Pardeshi, Jul 18 2025

			0.3015028323958245706837155695304698431433590983171469051779540518921...

Wikipedia, Covariance matrix.
Wikipedia, Regular icosahedron.
Wolfram Alpha, Johnson solid 2
Index entries for algebraic numbers, degree 2

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452.

Mathematica
```
RealDigits[N[(5+Sqrt[5])/24,200]]
```

(5+sqrt(5))/24 \\ Charles R Greathouse IV, Oct 30 2023

Equals (5+sqrt(5))/24.

A020761 Decimal expansion of 1/2.

Original entry on oeis.org

5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Real part of all nontrivial zeros of the Riemann zeta function (assuming the Riemann hypothesis to be true). - Alonso del Arte, Jul 02 2011
Radius of a sphere with surface area Pi. - Omar E. Pol, Aug 09 2012
Radius of the midsphere (tangent to the edges) in a regular octahedron with unit edges. Also radius of the inscribed sphere (tangent to faces) in a cube with unit edges. - Stanislav Sykora, Mar 27 2014
Construct a rectangle of maximal area inside an arbitrary triangle. The ratio of the rectangle's area to the triangle's area is 1/2. - Rick L. Shepherd, Jul 30 2014

			1/2 = 0.50000000000000...

Michael Penn, A creative approach to a scary looking integral., YouTube video, 2020.
Michael Penn, I really like this sum!, YouTube video, 2021.
Wikipedia, Riemann zeta function
Wikipedia, Platonic solid
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. In platonic solids:
midsphere radii:
A020765 (tetrahedron),
A010503 (cube),
A019863 (icosahedron),
A239798 (dodecahedron);
insphere radii:
A020781 (tetrahedron),
A020763 (octahedron),
A179294 (icosahedron),
A237603 (dodecahedron).

Digits:=100; evalf(1/2); # Wesley Ivan Hurt, Mar 27 2014

RealDigits[1/2, 10, 128][[1]] (* Alonso del Arte, Dec 13 2013 *)
LinearRecurrence[{1},{5,0},99] (* Ray Chandler, Jul 15 2015 *)

{ default(realprecision); x=1/2*10; for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Felix Fröhlich, Jul 24 2014

a(n) = 5*(n==0); \\ Michel Marcus, Jul 25 2014

Equals Sum_{k>=1} (1/3^k). Hence 1/2 = 0.1111111111111... in base 3.
Cosine of 60 degrees, i.e., cos(Pi/3).
-zeta(0), zeta being the Riemann function. - Stanislav Sykora, Mar 27 2014
a(0) = 5; a(n) = 0, n > 0. - Wesley Ivan Hurt, Mar 27 2014
a(n) = 5 * floor(1/(n + 1)). - Wesley Ivan Hurt, Mar 27 2014
Equals 2*A019824*A019884. - R. J. Mathar, Jan 17 2021

A179449 Decimal expansion of the volume of great icosahedron with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

Great icosahedron: 20 faces, 12 vertices, and 30 edges.

			0.31830500937508762649617763802863490189974235016186428155379281441228294...

Eric Weisstein's World of Mathematics, Great Icosahedron
Wikipedia, Great Icosahedron
Index entries for algebraic numbers, degree 2

Cf. A010527, A102208, A179290, A179292, A179294.

RealDigits[N[5*(Sqrt[5]-3)/12, 105]][[1]]

Digits of 5/12 * (3-sqrt(5)).

Partially rewritten by Charles R Greathouse IV, Feb 02 2011

A179451 Decimal expansion of the surface area of an icosidodecahedron with side length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.

			29.3059828449119895407453754361926770276025163091742830907641713815460...

Index entries for algebraic numbers, degree 8

Cf. A010527, A102208, A179290, A179292, A179294, A179449, A179450.

RealDigits[N[Sqrt[30*(10+3*Sqrt[5]+Sqrt[75+30*Sqrt[5]])],200]]

polrootsreal(x^8 - 1200*x^6 + 324000*x^4 - 27000000*x^2 + 324000000)[8] \\ Charles R Greathouse IV, Oct 30 2023

Sqrt(30*(10+3*sqrt(5)+sqrt(75+30*sqrt(5))))

Partially rewritten by Charles R Greathouse IV, Feb 03 2011

A179450 Decimal expansion of the volume of an icosidodecahedron with edge length 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 14 2010

Icosidodecahedron: 32 faces, 30 vertices, and 60 edges.

			13.83552593624940413982599206140528266708175201889932288543420886199647...

Index entries for algebraic numbers, degree 2

Cf. A010527, A102208, A179290, A179292, A179294, A179449.

Mathematica
```
RealDigits[N[(45+17*Sqrt[5])/6,200]]
```

(45 + 17*sqrt(5))/6 \\ Charles R Greathouse IV, Oct 30 2023

(45 + 17*sqrt(5))/6.

Partially rewritten by Charles R Greathouse IV, Feb 03 2011

A179590 Decimal expansion of the volume of pentagonal cupola with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

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

			2.32404531833319313093944911248751749029374557307435048284726483027368...

Wolfram Alpha, Johnson solid 5
Index entries for algebraic numbers, degree 2

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

Mathematica
```
RealDigits[N[(5+4*Sqrt[5])/6,200]]
```

Digits of (5+4*sqrt(5))/6.

A179553 Decimal expansion of the surface area of pentagonal pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Pentagonal pyramid: 6 faces, 6 vertices, and 10 edges.

			3.885540910050063539668319904270950058085880737273174114276853431338785...

Wolfram Alpha, Johnson solid 2
Index entries for algebraic numbers, degree 8

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

RealDigits[N[Sqrt[5/2*(10+Sqrt[5]+Sqrt[75+30*Sqrt[5]])]/2,200]]

Digits of sqrt(5/2*(10+sqrt(5)+sqrt(75+30sqrt(5))))/2.

cp's OEIS Frontend

A019881 Decimal expansion of sin(2*Pi/5) (sine of 72 degrees).

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A179587 Decimal expansion of the volume of square cupola with edge length 1.

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A179452 Decimal expansion of sqrt(5 + 2*sqrt(5))/2, the height of a regular pentagon and midradius of an icosidodecahedron with side length 1.

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A179552 Decimal expansion of the volume of pentagonal pyramid with edge length 1.

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A020761 Decimal expansion of 1/2.

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A179449 Decimal expansion of the volume of great icosahedron with edge length 1.

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