A379890 -id:A379890 - cp's OEIS frontend

A379888 Decimal expansion of the surface area of a pentagonal hexecontahedron with unit shorter edge length.

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

Offset: 3

Paolo Xausa, Jan 07 2025

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

			162.69896419846662676872582412137959709718223664038...

Paolo Xausa, Table of n, a(n) for n = 3..10000
Eric Weisstein's World of Mathematics, Pentagonal Hexecontahedron.
Wikipedia, Pentagonal hexecontahedron.
Index entries for algebraic numbers, degree 12.

Cf. A379889 (volume), A379890 (inradius), A379891 (midradius), A379892 (dihedral angle).
Cf. A377804 (surface area of a snub dodecahedron with unit edge length).
Cf. A001622.

First[RealDigits[Root[961*#^12 - 33925050*#^10 + 238487439375*#^8 - 374285139187500*#^6 + 215543322643359375*#^4 - 200764566730722656250*#^2 + 19088214930090087890625 &, 8], 10, 100]]  (* or *)
First[RealDigits[PolyhedronData["PentagonalHexecontahedron", "SurfaceArea"], 10, 100]]

Equals 30*(2 + 3*t)*sqrt(1 - t^2)/(1 - 2*t^2), 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 961*x^12 - 33925050*x^10 + 238487439375*x^8 - 374285139187500*x^6 + 215543322643359375*x^4 - 200764566730722656250*x^2 + 19088214930090087890625.

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

Paolo Xausa, Jan 07 2025

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

			189.78985206688527910632308619447379699106033629736...

Paolo Xausa, Table of n, a(n) for n = 3..10000
Eric Weisstein's World of Mathematics, Pentagonal Hexecontahedron.
Wikipedia, Pentagonal hexecontahedron.
Index entries for algebraic numbers, degree 12.

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

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]]

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.

A379892 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a pentagonal hexecontahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jan 10 2025

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

			2.6734732271767846682790703348957917197870317502693...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentagonal Hexecontahedron.
Wikipedia, Pentagonal hexecontahedron.
Index entries for transcendental numbers.

Cf. A379888 (surface area), A379889 (volume), A379890 (inradius), A379891 (midradius).
Cf. A377997 and A377998 (dihedral angles of a snub dodecahedron).
Cf. A377849.

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

acos(polrootsreal(209*x^6 - 94*x^5 - 137*x^4 + 100*x^3 - 9*x^2 - 6*x + 1)[1]) \\ Charles R Greathouse IV, Feb 10 2025

Equals arccos(A377849/(A377849 - 2)).

Equals arccos(c), where c is the smallest real root of 209*x^6 - 94*x^5 - 137*x^4 + 100*x^3 - 9*x^2 - 6*x + 1.

A379891 Decimal expansion of the midradius of a pentagonal hexecontahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jan 09 2025

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

			3.59762482255118901142825655944442353841196452...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentagonal Hexecontahedron.
Wikipedia, Pentagonal hexecontahedron.

Cf. A379888 (surface area), A379889 (volume), A379890 (inradius), A379892 (dihedral angle).

Cf. A377807 (midradius of a snub dodecahedron with unit edge length).

First[RealDigits[Root[4096*#^12 - 58368*#^10 + 70656*#^8 - 17728*#^6 + 1392*#^4 - 120*#^2 + 1 &, 8], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["PentagonalHexecontahedron", "Midradius"], 10, 100]]

Equals the largest real root of 4096*x^12 - 58368*x^10 + 70656*x^8 - 17728*x^6 + 1392*x^4 - 120*x^2 + 1.

A380002 Decimal expansion of long/short edge length ratio of a pentagonal hexecontahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jan 11 2025

			1.749852566736202791676446693655921179649815818590...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentagonal Hexecontahedron.
Wikipedia, Pentagonal hexecontahedron.

Cf. A379888 (surface area), A379889 (volume), A379890 (inradius), A379890 (midradius), A379892 (dihedral angle), A380003 and A380004 (face internal angles).

Cf. A377849.

First[RealDigits[(1 + #)/(2 - #^2) & [Root[#^3 + 2*#^2 - GoldenRatio^2 &, 1]], 10, 100]] (* or *)
First[RealDigits[1/Divide @@ PolyhedronData["PentagonalHexecontahedron", "EdgeLengths"], 10, 100]]

Equals (1 + xi)/(2 - xi^2), where xi = A377849.

Equals the largest real root of 31*x^6 - 122*x^5 + 177*x^4 - 128*x^3 + 51*x^2 - 11*x + 1.

A380003 Decimal expansion of acute vertex angle, in radians, in a pentagonal hexecontahedron face.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jan 12 2025

A pentagonal hexecontahedron face is an irregular pentagon with one acute angle (this constant) and four (equal) obtuse angles (A380004).

			1.1772858234717502919235374454812446809073054345981...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentagonal Hexecontahedron.
Wikipedia, Pentagonal hexecontahedron.

Cf. A379888 (surface area), A379889 (volume), A379890 (inradius), A379890 (midradius), A379892 (dihedral angle), A380002 (long/short edge length ratio), A380004 (face obtuse angles).

First[RealDigits[ArcCos[Root[64*#^6 - 384*#^5 + 384*#^4 + 888*#^3 + 168*#^2 - 128*# - 31 &, 4]], 10, 100]]

Equals arccos(c), where c is the largest real root of 64*x^6 - 384*x^5 + 384*x^4 + 888*x^3 + 168*x^2 - 128*x - 31.

Equals 3*Pi - 4*A380004.

A380004 Decimal expansion of obtuse vertex angles, in radians, in a pentagonal hexecontahedron face.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jan 12 2025

A pentagonal hexecontahedron face is an irregular pentagon with one acute angle (A380003) and four (equal) obtuse angles (this constant).

			2.06187303432440735586609817608931599292105069088...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentagonal Hexecontahedron.
Wikipedia, Pentagonal hexecontahedron.

Cf. A379888 (surface area), A379889 (volume), A379890 (inradius), A379890 (midradius), A379892 (dihedral angle), A380002 (long/short edge length ratio), A380003 (face acute angle).

First[RealDigits[ArcCos[Root[64*#^6 - 128*#^5 + 64*#^4 + 24*#^3 - 24*#^2 + 1 &, 1]], 10, 100]]

Equals arccos(c), where c is the smallest real root of 64*x^6 - 128*x^5 + 64*x^4 + 24*x^3 - 24*x^2 + 1.

Equals (3*Pi - A380003)/4.

cp's OEIS Frontend

A379888 Decimal expansion of the surface area of a pentagonal hexecontahedron with unit shorter edge length.

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

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A379892 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a pentagonal hexecontahedron.

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A379891 Decimal expansion of the midradius of a pentagonal hexecontahedron with unit shorter edge length.

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A380002 Decimal expansion of long/short edge length ratio of a pentagonal hexecontahedron.

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A380003 Decimal expansion of acute vertex angle, in radians, in a pentagonal hexecontahedron face.

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A380004 Decimal expansion of obtuse vertex angles, in radians, in a pentagonal hexecontahedron face.

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