A002163 -id:A002163 - cp's OEIS frontend

A377696 Decimal expansion of the circumradius of a truncated dodecahedron with unit edge length.

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

Offset: 1

Paolo Xausa, Nov 04 2024

			2.9694490158633984670421666956925979636007477003...

Eric Weisstein's World of Mathematics, Truncated Dodecahedron.
Wikipedia, Truncated dodecahedron.

Cf. A377694 (surface area), A377695 (volume), A377697 (midradius), A377698 (Dehn invariant, negated).
Cf. A179296 (analogous for a regular dodecahedron).
Cf. A002163.

First[RealDigits[Sqrt[74 + 30*Sqrt[5]]/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedDodecahedron", "Circumradius"], 10, 100]]

Equals sqrt(74 + 30*sqrt(5))/4 = sqrt(74 + 30*A002163)/4.

A378388 Decimal expansion of the surface area of a tetrakis hexahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 27 2024

The tetrakis hexahedron is the dual polyhedron of the truncated octahedron.

			11.925695879998878380848926233233473255683297917928...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Tetrakis Hexahedron.

Cf. A374359 (volume - 1), A010532 (inradius*10), A179587 (midradius + 1), A378389 (dihedral angle).
Cf. A377341 (surface area of a truncated octahedron with unit edge).
Cf. A002163, A208899.

First[RealDigits[16*Sqrt[5]/3, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TetrakisHexahedron", "SurfaceArea"], 10, 100]]

Equals (16/3)*sqrt(5) = (16/3)*A002163 = 16*A208899.

A380861 Decimal expansion of the smallest acute vertex angle, in radians, in a deltoidal hexecontahedron face.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Feb 06 2025

A deltoidal hexecontahedron face is a kite with one smallest acute angle (this constant), two largest acute angles (A380862) and one obtuse angle (A380863).

			1.1830367284200834147901361867998878650548206683684...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Deltoidal hexecontahedron.

Cf. A002163, A379385, A379386, A379387, A379388, A379389, A380862, A380863.

First[RealDigits[ArcCos[(9*Sqrt[5] - 5)/40], 10, 100]]

Equals arccos((9*sqrt(5) - 5)/40) = arccos((9*A002163 - 5)/40).

Equals 2*Pi - 2*A380862 - A380863.

A380863 Decimal expansion of the obtuse vertex angle, in radians, in a deltoidal hexecontahedron face.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Feb 07 2025

A deltoidal hexecontahedron face is a kite with one smallest acute angle (A380861), two largest acute angles (A380862) and one obtuse angle (this constant).

			2.0641778238390721349307678649869730069970513653...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Deltoidal hexecontahedron.

Cf. A002163, A379385, A379386, A379387, A379388, A379389, A380861, A380862.

First[RealDigits[ArcCos[(-5 - 2*Sqrt[5])/20], 10, 100]]

Equals arccos((-5 - 2*sqrt(5))/20) = arccos((-5 - 2*A002163)/20).

Equals 2*Pi - A380861 - 2*A380862.

A384909 Decimal expansion of the volume of an elongated pentagonal orthobicupola with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jun 12 2025

The elongated pentagonal orthobicupola is Johnson solid J_38.

Also the volume of an elongated pentagonal gyrobicupola (Johnson solid J_39) with unit edge.

			12.342299479604519768304624665067309540604246504993...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Elongated pentagonal gyrobicupola.
Wikipedia, Elongated pentagonal orthobicupola.

Cf. A384625 (surface area - 10).

Cf. A002163, A010476, A384283, A384624, A384871.

First[RealDigits[(10 + 8*Sqrt[5] + 15*Sqrt[5 + Sqrt[20]])/6, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J38", "Volume"], 10, 100]]

Equals (10 + 8*sqrt(5) + 15*sqrt(5 + 2*sqrt(5)))/6 = (10 + 8*A002163 + 15*sqrt(5 + A010476))/6.

Equals the largest root of 1296*x^4 - 8640*x^3 - 82440*x^2 - 109200*x + 76525.

A384910 Decimal expansion of the volume of an elongated pentagonal orthocupolarotunda with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jun 13 2025

The elongated pentagonal orthocupolarotunda is Johnson solid J_40.

Also the volume of an elongated pentagonal gyrocupolarotunda (Johnson solid J_41) with unit edge.

			16.936017129396028707278171583282433383851376941...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Elongated pentagonal gyrocupolarotunda.
Wikipedia, Elongated pentagonal orthocupolarotunda.

Cf. A384911 (surface area).

Cf. A002163, A010476, A384283, A384285, A384624, A384871, A384909.

First[RealDigits[5/12*(11 + 5*Sqrt[5] + 6*Sqrt[5 + Sqrt[20]]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J40", "Volume"], 10, 100]]

Equals (5/12)*(11 + 5*sqrt(5) + 6*sqrt(5 + 2*sqrt(5))) = (5/12)*(11 + 5*A002163 + 6*sqrt(5 + A010476)).

Equals the largest root of 1296*x^4 - 23760*x^3 + 26100*x^2 + 84000*x - 111875.

A385262 Decimal expansion of the volume of a gyroelongated pentagonal cupolarotunda with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jun 27 2025

The gyroelongated pentagonal cupolarotunda is Johnson solid J_47.

			15.991096162004890063062980011720804055694009940053...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Gyroelongated pentagonal cupolarotunda.

Cf. A385263 (surface area).

Cf. A002163, A384283, A385256, A385258, A385260.

First[RealDigits[5/12*(11 + 5*# + 2*Sqrt[2*(Sqrt[650 + 290*#] - # - 1)]) & [Sqrt[5]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J47", "Volume"], 10, 100]]

Equals (5/12)*(11 + 5*sqrt(5) + 2*sqrt(2*(sqrt(650 + 290*sqrt(5)) - sqrt(5) - 1))) = (5/12)*(11 + 5*A002163 + 2*sqrt(2*(sqrt(650 + 290*A002163) - A002163 - 1))).

Equals the largest real root of 1679616*x^8 - 61585920*x^7 + 851472000*x^6 - 5108832000*x^5 + 4745790000*x^4 + 21346200000*x^3 - 29019375000*x^2 - 4576875000*x - 405859375.

A385263 Decimal expansion of the surface area of a gyroelongated pentagonal cupolarotunda with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jun 30 2025

The gyroelongated pentagonal cupolarotunda is Johnson solid J_47.

			32.198786370350444777678239329896650406601165160912...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Gyroelongated pentagonal cupolarotunda.

Cf. A385262 (volume).

Cf. A002163, A002194, A384284, A385257, A385259, A385261.

First[RealDigits[5 + (35*Sqrt[3] + 7*Sqrt[25 + 10*Sqrt[5]])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J47", "SurfaceArea"], 10, 100]]

Equals 5 + (35*sqrt(3) + 7*sqrt(25 + 10*sqrt(5)))/4 = 5 + (35*A002194 + 7*sqrt(25 + 10*A002163))/4.

Equals the largest root of 256*x^8 - 10240*x^7 - 134400*x^6 + 7616000*x^5 - 756000*x^4 - 1373680000*x^3 + 2724312500*x^2 + 55840875000*x - 106054671875.

A385510 Decimal expansion of the surface area of an augmented pentagonal prism with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 02 2025

The augmented pentagonal prism is Johnson solid J_52.

			9.17300560874681113904547029628309156575755359383...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Augmented pentagonal prism.

Cf. A385509 (volume).

Cf. A002163, A002194.

First[RealDigits[4 + Sqrt[3] + Sqrt[25/4 + 5*Sqrt[5]/2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J52", "SurfaceArea"], 10, 100]]

Equals 4 + sqrt(3) + sqrt(25/4 + 5*sqrt(5)/2) = 4 + A002194 + sqrt(25/4 + 5*A002163/2).

Equals the largest root of 256*x^8 - 8192*x^7 + 105216*x^6 - 690176*x^5 + 2391264*x^4 - 3788288*x^3 + 193904*x^2 + 6427776*x - 5201159.

A385535 Decimal expansion of the surface area of a biaugmented pentagonal prism with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 03 2025

The biaugmented pentagonal prism is Johnson solid J_53.

			9.905056416315688432572916637788963932700358847641...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Biaugmented pentagonal prism.

Cf. A385534 (volume).

Cf. A002163, A010469, A385510.

First[RealDigits[3 + Sqrt[12] + Sqrt[25/4 + 5*Sqrt[5]/2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J53", "SurfaceArea"], 10, 100]]

Equals 3 + 2*sqrt(3) + sqrt(25/4 + 5*sqrt(5)/2) = 3 + A010469 + sqrt(25/4 + 5*A002163/2).

Equals the largest root of 256*x^8 - 6144*x^7 + 45824*x^6 - 50688*x^5 - 729376*x^4 + 2504064*x^3 - 583184*x^2 - 4746912*x + 1628761.

cp's OEIS Frontend

A377696 Decimal expansion of the circumradius of a truncated dodecahedron with unit edge length.

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A378388 Decimal expansion of the surface area of a tetrakis hexahedron with unit shorter edge length.

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A380861 Decimal expansion of the smallest acute vertex angle, in radians, in a deltoidal hexecontahedron face.

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A380863 Decimal expansion of the obtuse vertex angle, in radians, in a deltoidal hexecontahedron face.

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A384909 Decimal expansion of the volume of an elongated pentagonal orthobicupola with unit edge.

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A384910 Decimal expansion of the volume of an elongated pentagonal orthocupolarotunda with unit edge.

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A385262 Decimal expansion of the volume of a gyroelongated pentagonal cupolarotunda with unit edge.

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A385263 Decimal expansion of the surface area of a gyroelongated pentagonal cupolarotunda with unit edge.

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