A102208 -id:A102208 - cp's OEIS frontend

A179637 Decimal expansion of the surface area of pentagonal rotunda with edge length 1.

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Pentagonal rotunda: 20 vertices, 35 edges, and 17 faces.

			22.3472002653941282767984141581886130738180135134316226129799763167102...

Wolfram Alpha, Johnson solid 6

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449-A179452, A179552-A179554, A179587-A179593.

RealDigits[N[Sqrt[5*(145+58*Sqrt[5]+2*Sqrt[30*(65+29*Sqrt[5])])]/2,200]]

Digits of sqrt(5*(145+58*sqrt(5)+2*sqrt(30*(65+29*sqrt(5)))))/2.

Offset corrected by R. J. Mathar, Aug 15 2010

A185093 Decimal expansion of the volume of small rhombicosidodecahedron with edge = 1.

Original entry on oeis.org

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

Offset: 2

Vladimir Joseph Stephan Orlovsky, Feb 04 2012

Small Rhombicosidodecahedron: 62 faces, 60 vertices, and 120 edges.
Surface Area = 30+sqrt(30*(10+3*sqrt(5)+sqrt(75+30*sqrt(5)))) = 59.30598284491...
Circumradius = sqrt(11+4*sqrt(5))/2 = 2.23295050941569004950041538324968277293...
Midradius = sqrt(10+4*sqrt(5))/2 = 2.17625089948282151110005286599776788019807...
Quadratic number with denominator 3 and minimal polynomial 9x^2 - 360x - 605. - Charles R Greathouse IV, Apr 25 2016

			41.6153237824979670652886787977356702759259774762447486679520...

G. C. Greubel, Table of n, a(n) for n = 2..5001 [Offset adapted by _Georg Fischer_, Oct 18 2021]

Cf. A102208, A179290, A179292.

RealDigits[N[20 + (29*Sqrt[5])/3, 200]][[1]]
RealDigits[PolyhedronData["Rhombicosidodecahedron","Volume"],10,100][[1]] (* Harvey P. Dale, Aug 04 2025 *)

29*sqrt(5)/3+20 \\ Charles R Greathouse IV, Oct 01 2012

Offset changed by Georg Fischer, Jul 29 2021

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

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.

A179638 Decimal expansion of the volume of gyroelongated square pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Gyroelongated square pyramid: 9 vertices, 20 edges, and 13 faces.

			1.19270224223223255906019842837725158155262551828862015707793142188822...

Wolfram Alpha, Johnson solid 10

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A179554, A179587, A179588, A179589, A179590, A179591, A179592, A179593, A179637.

RealDigits[N[(Sqrt[2]+2*Sqrt[4+3*Sqrt[2]])/6,200]]

Digits of (sqrt(2)+2 sqrt(4+3 sqrt(2)))/6.

A071402 Rounded volume of a regular icosahedron with edge length n.

Original entry on oeis.org

0, 2, 17, 59, 140, 273, 471, 748, 1117, 1590, 2182, 2904, 3770, 4793, 5987, 7363, 8936, 10719, 12724, 14964, 17454, 20205, 23231, 26545, 30160, 34089, 38345, 42942, 47893, 53209, 58906, 64995, 71490, 78404, 85749, 93540, 101789, 110509

Offset: 0

Rick L. Shepherd, May 29 2002

The printed reference given shows in a table on p. 10 that Volume is "2.18170a^3" (a is edge). Both PARI (see Example here) and a handheld calculator show that 2.18169 is correct for a 5-decimal-place approximation.

			a(6)=471 because round(6^3*(3 + sqrt(5))*5/12) = round(216*2.181694990...) = round(471.24...) = 471.

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, pp. 10-11.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Icosahedron

Cf. A000578 (cube), A071399 (tetrahedron), A071400 (octahedron), A071401 (dodecahedron), A071398 (total surface area of icosahedron).

Cf. A102208 ((3+Sqrt(5)) * 5/12).

[Round(n^3 * (3+Sqrt(5)) * 5/12): n in [0..50]]; // Vincenzo Librandi, May 21 2011

for(n=0,100,print1(round(n^3*(3+sqrt(5))*5/12),","))

a(n) = round(n^3 * (3+sqrt(5)) * 5/12).

A179593 Decimal expansion of the volume of pentagonal rotunda with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Pentagonal rotunda: 20 vertices, 35 edges, and 17 faces.

			6.91776296812470206991299603070264133354087600944966144271710443099823...

Wolfram Alpha, Johnson solid 6
Index entries for algebraic numbers, degree 2.

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587, A179588, A179589, A179590, A179591, A179592.

Mathematica
```
RealDigits[N[(45+17*Sqrt[5])/12,200]]
```

Digits of (45+17*sqrt(5))/12.

A179592 Decimal expansion of the circumradius of pentagonal cupola with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

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

			2.232950509415690049500415383249682772934080730579181647457441260...

Wolfram Alpha, Johnson solid 5
Index entries for algebraic numbers, degree 4.

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587, A179588, A179589, A179590, A179591.

Mathematica
```
RealDigits[N[Sqrt[11+4*Sqrt[5]]/2,200]]
```

Digits of sqrt(11+4*sqrt(5))/2.

A377751 Decimal expansion of the volume of a truncated icosahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 07 2024

			55.28773075812273923639861693886121953098664736582...

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

Cf. A377750 (surface area), A377752 (circumradius), A205769 (midradius + 1), A377787 (Dehn invariant).
Cf. A102208 (analogous for a regular icosahedron).
Cf. A002163.

First[RealDigits[(125 + 43*Sqrt[5])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedIcosahedron", "Volume"], 10, 100]]

(125 + 43*sqrt(5))/4 \\ Charles R Greathouse IV, Feb 05 2025

Equals (125 + 43*sqrt(5))/4 = (125 + 43*A002163)/4.

A386000 Decimal expansion of the volume of a tridiminished icosahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 14 2025

The tridiminished icosahedron is Johnson solid J_63.

			1.277186493437438661452675653379955568670180...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Tridiminished icosahedron.

Cf. A386001 (surfacea area).

Cf. A002163, A020829, A102208, A179552, A385857, A386002.

First[RealDigits[5/8 + 7*Sqrt[5]/24, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J63", "Volume"], 10, 100]]

Equals 5/8 + 7*sqrt(5)/24 = 5/8 + 7*A002163/24.
Equals A102208 - 3*A179552 = A386002 - A020829.
Equals the largest root of 144*x^2 - 180*x - 5.

cp's OEIS Frontend

A179637 Decimal expansion of the surface area of pentagonal rotunda with edge length 1.

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A185093 Decimal expansion of the volume of small rhombicosidodecahedron with edge = 1.

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PARI

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

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

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A179638 Decimal expansion of the volume of gyroelongated square pyramid with edge length 1.

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A071402 Rounded volume of a regular icosahedron with edge length n.

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Magma

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A179593 Decimal expansion of the volume of pentagonal rotunda with edge length 1.

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