A010527 -id:A010527 - cp's OEIS frontend

A339262 Decimal expansion of the conjecturally maximum possible volume of a polyhedron with 10 vertices inscribed in the unit sphere.

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

Offset: 1

Hugo Pfoertner, Dec 07 2020

The polyhedron (see linked illustration) has vertices at the poles and two square rings of vertices rotated by Pi/4 against each other, with a polar angle of approx. +-62.89908285 degrees against the poles. The polyhedron is completely described by this angle and its order 16 symmetry. It would be desirable to know a closed formula representation of this angle and the volume.

			2.218711131545399403247282751128417013810725374663344381750049084201...

R. H. Hardin, N. J. A. Sloane, and W. D. Smith, Maximal Volume Spherical Codes.
Hugo Pfoertner, Visualization of Polyhedron, (1999).
Hugo Pfoertner, Number of edges incident with the 10 vertices, video (2021).

Cf. A010527 (volume of double 5-pyramid), A081314, A081366, A122553 (volume of octahedron), A339259, A339260, A339261, A339263.

A377750 Decimal expansion of the surface area of a truncated icosahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 06 2024

			72.60725303413392187893153397383948620117264765443...

Eric Weisstein's World of Mathematics, Truncated Icosahedron.
Wikipedia, Truncated icosahedron.
Index entries for algebraic numbers, degree 8.

Cf. A377751 (volume), A377752 (circumradius), A205769 (midradius + 1), A377787 (Dehn invariant).
Cf. A010527 (analogous for a regular icosahedron, with offset 1).
Cf. A002163, A002194, A375067.

First[RealDigits[3*(10*Sqrt[3] + Sqrt[25 + Sqrt[500]]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedIcosahedron", "SurfaceArea"], 10, 100]]

3*(10*sqrt(3) + sqrt(25 + 10*sqrt(5))) \\ Charles R Greathouse IV, Feb 05 2025

Equals 3*(10*sqrt(3) + sqrt(25 + 10*sqrt(5))) = 30*A002194 + 3*sqrt(25 + 10*A002163).

Equals 30*(A002194 + A375067).

A386412 Decimal expansion of the surface area of an augmented truncated tetrahedron with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 21 2025

The augmented truncated tetrahedron is Johnson solid J_65.

			14.258330249197702407928401219788170385128234149767...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Augmented truncated tetrahedron.

Cf. A386411 (volume).

Cf. A010527.

First[RealDigits[3 + 13*Sqrt[3]/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J65", "SurfaceArea"], 10, 100]]

Equals 3 + 13*sqrt(3)/2 = 3 + 13*A010527.

Equals the largest root of 4*x^2 - 24*x - 471.

A126664 Continued fraction expansion of sqrt(3)/2.

Original entry on oeis.org

0, 1, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6

Offset: 0

Zerinvary Lajos, Feb 10 2007

Also continued fraction expansion of sin(Pi/3).

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010527.

ContinuedFraction[Sqrt[3]/2,120] (* or *) PadRight[{0,1},120,{6,2}] (* Harvey P. Dale, May 14 2016 *)

a(n) = 2*(2+(-1)^n) for n>1. a(n) = a(n-2) for n>3. G.f.: -x*(x^2+6*x+1) / ((x-1)*(x+1)). - Colin Barker, Jan 10 2014

More terms from N. J. A. Sloane, Dec 29 2008

A152422 Decimal expansion of (sqrt(3)-1)/2.

Original entry on oeis.org

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

Offset: 0

Geoffrey Caveney, Dec 03 2008

The number has continued fraction [0, 2, 1, 2, 1, 2, 1, ...].

The iterated function z^2 - 1/2 gives a good rational approximation of this number times -1 after sixty steps. - Alonso del Arte, Apr 10 2016

			0.36602540378443864676372317...

A variant of A010527, which is the main entry. - N. J. A. Sloane, Dec 04 2008

Digits:=100; evalf((sqrt(3)-1)/2); # Wesley Ivan Hurt, Apr 18 2014

RealDigits[(Sqrt[3] - 1)/2, 10, 100] [[1]] (* Wesley Ivan Hurt, Apr 18 2014 *)

(sqrt(3)-1)/2 \\ Altug Alkan, Apr 11 2016

Equals Product_{k>=1} (1 + (-1)^k * 2/(6*k-3)). - Amiram Eldar, Aug 10 2020

a(98) corrected by Georg Fischer, Apr 03 2020

A179639 Decimal expansion of the volume of gyroelongated pentagonal pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Gyroelongated pentagonal pyramid: 11 vertices,25 edges,and 16 faces.

			1.88019215822908780282010679244089525495689855152098881326825313369561...

Wolfram Alpha, Johnson solid 11

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A179554, A179587, A179588, A179589, A179590, A179591, A179592, A179593, A179637, A179638.

Mathematica
```
RealDigits[N[(25+9*Sqrt[5])/24,200]]
```

Digits of (25+9*sqrt(5))/24.

A179640 Decimal expansion of the surface area of gyroelongated pentagonal pyramid with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Gyroelongated pentagonal pyramid: 11 vertices, 25 edges, and 16 faces.

			8.21566792897225677348693575803563097544289387179912568441637087996861...

Wolfram Alpha, Johnson solid 11

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A179554, A179587, A179588, A179589, A179590, A179591, A179592, A179593, A179637, A179638, A179639.

RealDigits[N[Sqrt[5/2*(70+Sqrt[5]+3*Sqrt[75+30*Sqrt[5]])]/2,200]]

Digits of sqrt(5/2*(70+sqrt(5)+3*sqrt(75+30*sqrt(5))))/2.

A179641 Decimal expansion of the volume of pentagonal dipyramid with edge length 1.

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

Pentagonal dipyramid: 7 vertices, 15 edges, and 10 faces.

			0.60300566479164914136743113906093968628671819663429381035590810378421...

Wolfram Alpha, Johnson solid 13

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449-A179452, A179552-A179554, A179587-A179593, A179637-A179640.

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

(sqrt(5)+5)/12 \\ Charles R Greathouse IV, Apr 25 2016

Digits of (5+sqrt(5))/12.

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

A339263 Decimal expansion of the conjecturally maximum possible volume of a polyhedron with 11 vertices inscribed in the unit sphere.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Dec 07 2020

The polyhedron (see linked illustration) with a symmetry group of order 4 has a vertex in the north pole on its axis of symmetry. The remaining 10 vertices are diametrically opposite in pairs relative to this axis of symmetry. The polar vertex has vertex degree 6. 8 vertices have vertex degree 5. 2 vertices have vertex degree 4.

This allocation seems to be the best possible approximation of a medial distribution of the vertex degrees, which is a known necessary condition for maximum volume. Of the 25 possible triangulations with vertex degree >= 4, all the others have more than 2 vertices with vertex degree 4, which leads to more pointed corners and therefore smaller volumes.

			2.35463449506861520323688059263892654160344864269342168599607566...

R. H. Hardin, N. J. A. Sloane and W. D. Smith, Maximal Volume Spherical Codes.
Hugo Pfoertner, Visualization of Polyhedron, (1999).
Hugo Pfoertner, Number of edges incident with the 11 vertices, video (2021).

Cf. A010527 (volume of double 5-pyramid), A081314, A081366, A122553 (volume of octahedron), A339259, A339260, A339261, A339262.

A377346 Decimal expansion of the midradius of a truncated cuboctahedron (great rhombicuboctahedron) with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Oct 26 2024

			2.26303343845371462359202580343253714222906720265...

Eric Weisstein's World of Mathematics, Great Rhombicuboctahedron.
Wikipedia, Truncated cuboctahedron.

Cf. A377343 (surface area), A377344 (volume), A377345 (circumradius).
Cf. A010527 (analogous for a cuboctahedron).
Cf. A010524, A230981.

First[RealDigits[Sqrt[3 + 3/Sqrt[2]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedCuboctahedron", "Midradius"], 10, 100]]

Equals sqrt(12 + 6*sqrt(2))/2 = sqrt(12 + A010524)/2 = sqrt(3 + 3/sqrt(2)) = sqrt(3 + A230981).

cp's OEIS Frontend

A339262 Decimal expansion of the conjecturally maximum possible volume of a polyhedron with 10 vertices inscribed in the unit sphere.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A377750 Decimal expansion of the surface area of a truncated icosahedron with unit edge length.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A386412 Decimal expansion of the surface area of an augmented truncated tetrahedron with unit edge.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A126664 Continued fraction expansion of sqrt(3)/2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A152422 Decimal expansion of (sqrt(3)-1)/2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A179639 Decimal expansion of the volume of gyroelongated pentagonal pyramid with edge length 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A179640 Decimal expansion of the surface area of gyroelongated pentagonal pyramid with edge length 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A179641 Decimal expansion of the volume of pentagonal dipyramid with edge length 1.

Views

Author