cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A379889 Decimal expansion of the volume of a pentagonal hexecontahedron with unit shorter edge length.

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Jan 07 2025

Keywords

Comments

The pentagonal hexecontahedron is the dual polyhedron of the snub dodecahedron.

Examples

			189.78985206688527910632308619447379699106033629736...

Links

Crossrefs

Cf. A379888 (surface area), A379890 (inradius), A379891 (midradius), A379892 (dihedral angle).
Cf. A377805 (volume of a snub dodecahedron with unit edge length).
Cf. A001622.

Programs

  • Mathematica
    First[RealDigits[Root[3936256*#^12 - 143719449600*#^10 + 69717538560000*#^8 - 965464153000000*#^6 - 5195593956250000*#^4 - 6093827421875000*#^2 + 171855712890625 &, 8], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["PentagonalHexecontahedron", "Volume"], 10, 100]]

Formula

Equals 5*(1 + t)*(2 + 3*t)/((1 - 2*t^2)*sqrt(1 - 2*t)), where t = ((44 + 12*A001622*(9 + sqrt(81*A001622 - 15)))^(1/3) + (44 + 12*A001622*(9 - sqrt(81*A001622 - 15)))^(1/3) - 4)/12.
Equals the largest real root of 3936256*x^12 - 143719449600*x^10 + 69717538560000*x^8 - 965464153000000*x^6 - 5195593956250000*x^4 - 6093827421875000*x^2 + 171855712890625.

A377804 Decimal expansion of the surface area of a snub dodecahedron with unit edge length.

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Nov 08 2024

Keywords

Examples

			55.2867449584451489436570705587807625317445951163...

Links

Crossrefs

Cf. A377805 (volume), A377806 (circumradius), A377807 (midradius).
Cf. A131595 (analogous for a regular dodecahedron).
Cf. A002194.

Programs

  • Mathematica
    First[RealDigits[20*Sqrt[3] + 3*Sqrt[25 + 10*Sqrt[5]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["SnubDodecahedron", "SurfaceArea"], 10, 100]]

Formula

Equals 20*sqrt(3) + 3*sqrt(25 + 10*sqrt(5)) = 20*A002194 + A131595.

A377807 Decimal expansion of the midradius of a snub dodecahedron with unit edge length.

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Nov 10 2024

Keywords

Examples

			2.0970538352520879924039590523482862400308397305810...

Links

Crossrefs

Cf. A377804 (surface area), A377805 (volume), A377806 (circumradius).
Cf. A239798 (analogous for a regular dodecahedron).
Cf. A377849.

Programs

  • Mathematica
    First[RealDigits[Sqrt[1/(1 - Root[#^3 + 2*#^2 - GoldenRatio^2 &, 1])]/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["SnubDodecahedron", "Midradius"], 10, 100]]

Formula

Equals sqrt(1/(1 - A377849))/2.
Equals the real root closest to 2 of 4096*x^12 - 21504*x^10 + 16384*x^8 - 4672*x^6 + 624*x^4 - 40*x^2 + 1.

A377849 Decimal expansion of the real root of x^3 + 2*x^2 - phi^2, where phi is the golden ratio (A001622).

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Nov 09 2024

Keywords

Comments

Ratio of the midradius of a snub dodecahedron to the midradius of the icosahedron in which it is inscribed (see Wikipedia article).

Examples

			0.94315125924388181712671989257036415940665038623453...

Links

Crossrefs

Cf. A001622, A377805, A377806, A377807.

Programs

  • Mathematica
    First[RealDigits[Root[#^3 + 2*#^2 - GoldenRatio^2 &, 1], 10,100]]
  • PARI
    polrootsreal(x^6 + 4*x^5 + 4*x^4 - 3*x^3 - 6*x^2 + 1)[4] \\ Charles R Greathouse IV, Feb 10 2025

Formula

Equals the real root closest to 1 of x^6 + 4*x^5 + 4*x^4 - 3*x^3 - 6*x^2 + 1.

A377806 Decimal expansion of the circumradius of a snub dodecahedron with unit edge length.

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Nov 10 2024

Keywords

Examples

			2.1558373751156397018366290766930582770168512187748...

Links

Crossrefs

Cf. A377804 (surface area), A377805 (volume), A377807 (midradius).
Cf. A179296 (analogous for a regular dodecahedron).
Cf. A377849.

Programs

  • Mathematica
    First[RealDigits[Sqrt[1 + 1/(1 - Root[#^3 + 2*#^2 - GoldenRatio^2 &, 1])]/2, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["SnubDodecahedron", "Circumradius"], 10, 100]]

Formula

Equals sqrt(1 + 1/(1 - A377849))/2.
Equals the real root closest to 2 of 4096*x^12 - 27648*x^10 + 47104*x^8 - 35776*x^6 + 13872*x^4 -2696*x^2 + 209.

A377997 Decimal expansion of the dihedral angle, in radians, between triangular faces in a snub dodecahedron.

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Nov 15 2024

Keywords

Examples

			2.8654006883447286076046073417336569141190096726652...

Links

Crossrefs

Cf. A377804, A377805, A377806, A377807, A377849, A377998.

Programs

  • Mathematica
    First[RealDigits[Pi - ArcCos[2*Root[#^3 + 2*#^2 - GoldenRatio^2 &, 1]/3 + 1/3], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["SnubDodecahedron", "DihedralAngles"]], 10, 100]]

Formula

Equals Pi - arccos((2/3)*A377849 + 1/3).
Equals Pi - arccos(c), where c is the largest real root of 729*x^6 + 486*x^5 - 729*x^4 - 756*x^3 + 63*x^2 + 270*x + 1.

A377998 Decimal expansion of the dihedral angle, in radians, between triangular and pentagonal faces in a snub dodecahedron.

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Nov 15 2024

Keywords

Examples

			2.669130633625756107707940935718208230518703745355...

Links

Crossrefs

Cf. A001622, A377804, A377805, A377806, A377807, A377849, A377997.

Programs

  • Mathematica
    First[RealDigits[Pi - ArcCos[Root[91125*#^12 - 668250*#^10 + 2006775*#^8 - 2735100*#^6 + 1768275*#^4 - 502410*#^2 + 43681&, 7]], 10, 100]] (* or *)
First[RealDigits[Min[PolyhedronData["SnubDodecahedron", "DihedralAngles"]], 10, 100]]

Formula

Equals Pi - arccos(sqrt((12*phi - (4*phi + 8)*xi*(xi + 1) + 19)/15)), where phi = A001622 and xi = A377849.
Equals Pi - arccos(c), where c is the real root closest to 9/10 of 91125*x^12 - 668250*x^10 + 2006775*x^8 - 2735100*x^6 + 1768275*x^4 - 502410*x^2 + 43681.

A381696 Decimal expansion of the isoperimetric quotient of a snub dodecahedron.

Original entry on oeis.org

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

Views

Author

Paolo Xausa, Mar 10 2025

Keywords

Comments

For the definition of isoperimetric quotient of a solid, references and links, see A381684.
The snub dodecahedron is the Archimedean solid with the highest isoperimetric quotient.

Examples

			0.94699904523421562618454412877087470550479673815077...

Links

Crossrefs

Cf. A377804 (surface area), A377805 (volume).
Cf. A000796, A002163, A002194, A381684.

Programs

  • Mathematica
    First[RealDigits[Pi/(20*Sqrt[3] + 3*Sqrt[25 + Sqrt[500]])^3*Root[#^6 - 52845*#^5 + 96583500*#^4 + 22087761875*#^3 + 4747626384375*#^2 - 5059009765293750*# + 187445810737515625 &, 4], 10, 100]]

Formula

Equals 36*Pi*A377805^2/(A377804^3).
Equals Pi/((20*sqrt(3) + 3*sqrt(25 + 10*sqrt(5)))^3)*r = A000796/((20*A002194 + 3*sqrt(25 + 10*A002163))^3)*r, where r is the largest real root of x^6 - 52845*x^5 + 96583500*x^4 + 22087761875*x^3 + 4747626384375*x^2 - 5059009765293750*x + 187445810737515625.
Showing 1-8 of 8 results.

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