A339259 -id:A339259 - cp's OEIS frontend

A122553 a(0)=1, a(n)=3 for n > 0.

1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 0

Philippe Deléham, Sep 20 2006

Continued fraction for (sqrt(13) - 1)/2 = A223139.
Decimal expansion of 4/30. - Alonso del Arte, Aug 16 2012
4/3 is the volume of the regular octahedron inscribed in the unit-radius sphere. - Amiram Eldar, Jun 02 2023

Calvin C. Clawson, Mathematical Mysteries, The Beauty and Magic of Numbers, Springer, 2013, pp. 95-96, 224.

Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A118273 (cube), A339259 (regular icosahedron), A363437 (regular tetrahedron), A363438 (regular dodecahedron).

Cf. A223139.

RealDigits[4/3, 10, 105][[1]] (* Alonso del Arte, Aug 16 2012 *)
PadRight[{1},120,3] (* Harvey P. Dale, Jul 21 2023 *)

a(n)=(n>=0)+2*(n>0) \\ Jaume Oliver Lafont, Mar 26 2009

a(n) = 3 - 2*0^n.
G.f.: (1 + 2*x)/(1 - x).
Sum_{n >= 0} a(n)*10^(-n) = 4/3.
From Amiram Eldar, Jun 05 2021: (Start)
4/3 = Product_{k>=1} (1 + 1/2^(2^k)).
4/3 = Sum_{k>=0} binomial(2*k,k)/((k+2)*4^k). (End)
Sum_{k>0} 3*k/4^k = 4/3 [Nicole Oresme]. - Stefano Spezia, Jun 27 2024
K_{n>=3} n/(n-2) = 4/3 (see Clawson at p. 224). - Stefano Spezia, Jul 01 2024
E.g.f.: 3*exp(x) - 2. - Elmo R. Oliveira, Aug 05 2024

A118273 Decimal expansion of (4/3)^(3/2).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Apr 21 2006

The volume of the cube inscribed in the unit-radius sphere. - Amiram Eldar, Jun 02 2023

			1.539600717839002038...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.24, p. 412.

Elliott H. Lieb, Residual entropy of square ice, Phys. Rev. 162 (1967) 162-172.
James Propp, The many faces of alternating-sign matrices, 2002.
Scenta server, Lieb's square ice constant. [Broken link]
Eric Weisstein's World of Mathematics, Lieb's Square Ice Constant.
Wikipedia, Lieb's square ice constant.
Index entries for algebraic numbers, degree 2.

Cf. A020784, A054759.

Cf. A122553 (octahedron), A339259 (regular icosahedron), A363437 (regular tetrahedron), A363438 (regular dodecahedron).

evalf(sqrt((4/3)^3)) ; # R. J. Mathar, Mar 29 2012

RealDigits[(4/3)^(3/2),10,120][[1]] (* Harvey P. Dale, Aug 06 2015 *)

(4/3)^(3/2) \\ Stefano Spezia, Dec 15 2024

Equals 8 * A020784.

A339260 Decimal expansion of the maximum possible volume of a polyhedron with 8 vertices inscribed in the unit sphere.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Nov 29 2020

Berman and Hanes (see link, page 81) proved in 1970 that an arrangement of 8 points on the surface of a sphere with 4 points with node degree 4 and 4 points with node degree 5 is the one with a maximum volume of their convex hull.

			1.8157161042244203975084949306331777890131009552754398376663729...

Joel D. Berman and Kitt Hanes, Volumes of Polyhedra Inscribed in the Unit Sphere in E3. Mathematische Annalen 188, 78-84 (1970).
Donald W. Grace, Search For Largest Polyhedra, Mathematics of Computation 17 (1963), pp. 197-199.
Matt Parker, The search for the biggest shape in the universe, YouTube video, 2024.
Hugo Pfoertner, Visualization of Polyhedron, (1999).
Hugo Pfoertner, Number of edges incident with the 8 vertices, video (2021).

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

RealDigits[Sqrt[(475 + 29*Sqrt[145])/250], 10, 120][[1]] (* Amiram Eldar, Jun 01 2023 *)

PARI
```
sqrt((475+29*sqrt(145))/250)
```

Equals sqrt((475 + 29*sqrt(145))/250).

A339261 Decimal expansion of the conjecturally maximum possible volume of a polyhedron with 9 vertices inscribed in the unit sphere.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Dec 05 2020

			2.0437501158996398411663654642269853338632606152947518187182157956871...

R. H. Hardin, N. J. A. Sloane and W. D. Smith, Maximal Volume Spherical Codes.
Hugo Pfoertner, Visualization of Polyhedron, (1999).
Hugo Pfoertner, 9-Vertex-Polyhedron with maximum volume inscribed in a sphere, YouTube video, Feb 10 2021.

Cf. A010527 (volume of double 5-pyramid), A081314, A081366, A122553 (volume of octahedron), A339259, A339260, A339261, A339262, A339263.

RealDigits[3*Sqrt[2*Sqrt[3] - 3], 10, 120][[1]] (* Amiram Eldar, Jun 28 2023 *)

PARI
```
3*sqrt(2*sqrt(3) - 3)
```

Equals 3*sqrt(2*sqrt(3) - 3).

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

Original entry on oeis.org

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.

A363437 Decimal expansion of the volume of the regular tetrahedron inscribed in the unit-radius sphere.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 02 2023

			0.51320023927966734623035447155729551613120155668455...

Eric Weisstein's World of Mathematics, Regular Tetrahedron.
Wikipedia, Tetrahedron: Regular tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. A118273 (cube), A122553 (regular octahedron), A339259 (regular icosahedron), A363438 (regular dodecahedron).

Other constants related to the regular tetrahedron: A020781, A020829, A137914, A156546, A187110, A210974, A232812, A236555.

Mathematica
```
RealDigits[8/(9*Sqrt[3]), 10, 120][[1]]
```

8/9/sqrt(3) \\ Charles R Greathouse IV, Feb 07 2025

Equals 8/(9*sqrt(3)).
Equals A118273 / 3.
Equals A020829 / A187110 ^ 3.

A363438 Decimal expansion of the volume of the regular dodecahedron inscribed in the unit-radius sphere.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 02 2023

			2.78516386312262296729255491273594698789932177207633...

Eric Weisstein's World of Mathematics, Regular Dodecahedron.
Wikipedia, Regular dodecahedron.
Index entries for algebraic numbers, degree 4.

Cf. A118273 (cube), A122553 (regular octahedron), A339259 (regular icosahedron), A363437 (regular tetrahedron).
Cf. A001622.
Other constants related to the regular dodecahedron: A102769, A131595, A179296, A232810, A237603, A239798, A341906.

RealDigits[(2*(5 + Sqrt[5]))/(3*Sqrt[3]), 10, 120][[1]]

2*sqrt(5+sqrt(5))/sqrt(27) \\ Charles R Greathouse IV, Feb 07 2025

Equals 2*sqrt(5+sqrt(5))/(3*sqrt(3)).
Equals 4*(phi+2)/(3*sqrt(3)), where phi is the golden ratio (A001622).
Equals A102769 / A179296 ^ 3.

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.

cp's OEIS Frontend

A122553 a(0)=1, a(n)=3 for n > 0.

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A118273 Decimal expansion of (4/3)^(3/2).

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A339260 Decimal expansion of the maximum possible volume of a polyhedron with 8 vertices inscribed in the unit sphere.

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A339261 Decimal expansion of the conjecturally maximum possible volume of a polyhedron with 9 vertices inscribed in the unit sphere.

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

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A363437 Decimal expansion of the volume of the regular tetrahedron inscribed in the unit-radius sphere.

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A363438 Decimal expansion of the volume of the regular dodecahedron inscribed in the unit-radius sphere.

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