A002163 -id:A002163 - cp's OEIS frontend

A378976 Decimal expansion of the midradius of a triakis icosahedron with unit shorter edge length.

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

Offset: 1

Paolo Xausa, Dec 14 2024

The triakis icosahedron is the dual polyhedron of the truncated dodecahedron.

			1.3944271909999158785636694674925104941762473438446...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triakis Icosahedron.
Wikipedia, Triakis icosahedron.

Cf. A378973 (surface area), A378974 (volume), A378975 (inradius), A378977 (dihedral angle).
Cf. A377697 (midradius of a truncated dodecahedron with unit edge length).
Cf. A002163, A010532, A249600.

First[RealDigits[1/2 + 2/Sqrt[5], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisIcosahedron", "Midradius"], 10, 100]]

Equals 1/2 + 2/sqrt(5) = 1/2 + 2/A002163.

Equals (A249600 + 13)/10 = (A010532 + 5)/10.

A378977 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a triakis icosahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 14 2024

The triakis icosahedron is the dual polyhedron of the truncated dodecahedron.

			2.8032178560848059621034493264877253281152659880354...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triakis Icosahedron.
Wikipedia, Triakis icosahedron.

Cf. A378973 (surface area), A378974 (volume), A378975 (inradius), A378976 (midradius).
Cf. A137218 and A344075 (dihedral angles of a truncated dodecahedron).
Cf. A002163.

First[RealDigits[ArcCos[-3*(8 + 5*Sqrt[5])/61], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["TriakisIcosahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-3*(8 + 5*sqrt(5))/61) = arccos(-3*(8 + 5*A002163)/61).

A379132 Decimal expansion of the surface area of a pentakis dodecahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Dec 16 2024

The pentakis dodecahedron is the dual polyhedron of the truncated icosahedron.

			27.93524960070079310581019127996368070525778610907...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Pentakis Dodecahedron.
Wikipedia, Pentakis dodecahedron.
Index entries for algebraic numbers, degree 4.

Cf. A379133 (volume), A379134 (inradius), A379135 (midradius), A379136 (dihedral angle).
Cf. A377750 (surface area of a truncated icosahedron with unit edge length).
Cf. A002163.

First[RealDigits[5/3*Sqrt[(421 + 63*Sqrt[5])/2], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["PentakisDodecahedron", "SurfaceArea"], 10, 100]]

sqrt((421 + 63*sqrt(5))/2)*5/3 \\ Charles R Greathouse IV, Feb 05 2025

Equals (5/3)*sqrt((421 + 63*sqrt(5))/2) = (5/3)*sqrt((421 + 63*A002163)/2).

A379133 Decimal expansion of the volume of a pentakis dodecahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Dec 16 2024

The pentakis dodecahedron is the dual polyhedron of the truncated icosahedron.

			13.458569366318714223642964127539153595279924859762...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Pentakis Dodecahedron.
Wikipedia, Pentakis dodecahedron.
Index entries for algebraic numbers, degree 2.

Cf. A379132 (surface area), A379134 (inradius), A379135 (midradius), A379136 (dihedral angle).
Cf. A377751 (volume of a truncated icosahedron with unit edge length).
Cf. A002163.

First[RealDigits[5/36*(41 + 25*Sqrt[5]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["PentakisDodecahedron", "Volume"], 10, 100]]

(41 + 25*sqrt(5))*5/36 \\ Charles R Greathouse IV, Feb 05 2025

Equals (5/36)*(41 + 25*sqrt(5)) = (5/36)*(41 + 25*A002163).

A379387 Decimal expansion of the inradius of a deltoidal hexecontahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 23 2024

The deltoidal hexecontahedron is the dual polyhedron of the (small) rhombicosidodecahedron.

			2.634797688222471365013793337475980265570278715884...

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

Cf. A379385 (surface area), A379386 (volume), A379388 (midradius), A379389 (dihedral angle).

Cf. A002163.

First[RealDigits[Root[820*#^4 - 5710*#^2 + 121 &, 4], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["DeltoidalHexecontahedron", "Inradius"], 10, 100]]

Equals 11*sqrt((135 + 59*sqrt(5))/205)/(7 - sqrt(5)) = 11*sqrt((135 + 59*A002163)/205)/(7 - A002163).

Equals the largest root of 820*x^4 - 5710*x^2 + 121.

A379389 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a deltoidal hexecontahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 23 2024

The deltoidal hexecontahedron is the dual polyhedron of the (small) rhombicosidodecahedron.

			2.6899252342065763400728815146316168300353303724921...

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

Cf. A379385 (surface area), A379386 (volume), A379387 (inradius), A379388 (midradius).
Cf. A377995 and A377996 (dihedral angles of a (small) rhombicosidodecahedron).
Cf. A002163.

First[RealDigits[ArcCos[-(19 + 8*Sqrt[5])/41], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["DeltoidalHexecontahedron", "DihedralAngles"]], 10, 100]]

Equals arccos(-(19 + 8*sqrt(5))/41) = arccos(-(19 + 8*A002163)/41).

A384138 Decimal expansion of the volume of an elongated pentagonal pyramid with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, May 20 2025

The elongated pentagonal pyramid is Johnson solid J_9.

			2.0219802329847914934427275469190794425507332683273...

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

Cf. A179553 (surface area - 5).

Cf. A002163, A383852, A384139, A384140.

First[RealDigits[(5 + Sqrt[5] + 6*Sqrt[25 + 10*Sqrt[5]])/24, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J9", "Volume"], 10, 100]]

Equals (5 + sqrt(5) + 6*sqrt(25 + 10*sqrt(5)))/24 = (5 + A002163 + 6*sqrt(25 + 10*A002163))/24.

Equals the largest root of 20736*x^4 - 17280*x^3 - 59760*x^2 + 15600*x + 9025.

A384140 Decimal expansion of the volume of an elongated pentagonal bipyramid with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, May 20 2025

The elongated pentagonal bipyramid is Johnson solid J_16.

			2.32348306538061606412644311644954928569409236664...

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

Cf. A384141 (surface area).

Cf. A002163, A383852, A384138, A384139.

First[RealDigits[(5 + Sqrt[5] + 3*Sqrt[25 + 10*Sqrt[5]])/12, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J16", "Volume"], 10, 100]]

Equals (5 + sqrt(5) + 3*sqrt(25 + 10*sqrt(5)))/12 = (5 + A002163 + 3*sqrt(25 + 10*A002163))/12.

Equals the largest root of 20736*x^4 - 34560*x^3 - 44640*x^2 + 27600*x + 6025.

A138367 Count of post-period decimal digits up to which the rounded n-th convergent to sqrt(5) agrees with the exact value.

Original entry on oeis.org

0, 2, 4, 5, 6, 7, 8, 10, 8, 12, 14, 14, 16, 18, 19, 20, 21, 23, 24, 24, 26, 28, 29, 30, 31, 33, 33, 34, 35, 37, 39, 40, 41, 42, 44, 44, 46, 47, 48, 49, 51, 53, 53, 55, 56, 57, 59, 60, 60, 61, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79, 80, 81, 83, 83, 85, 85, 88, 89, 90, 91, 92

Offset: 1

Artur Jasinski, Mar 17 2008

This is a measure of the quality of the n-th convergent to A002163 if the convergent and the exact value are compared rounded to an increasing number of digits.
The sequence of rounded values of sqrt(5) is 2, 2.2, 2.24, 2.236, 2.2361, 2.23607, 2.236068, 2.2360680 etc, and the n-th convergent (provided by A001077 and A001076) is to be represented by its equivalent sequence.
a(n) represents the maximum number of post-period digits of the two sequences if compared at the same level of rounding. Counting only post-period digits (which is one less than the full number of decimal digits) is just a convention taken from A084407.

			For n=3, the 3rd convergent is 161/72 = 2.236111111..., with a sequence of rounded representations 2, 2.2, 2.24, 2.236, 2.2361, 2.23611, 2.236111, 2.2361111 etc.
Rounded to 1, 2, 3, or 4 post-period decimal digits, this is the same as the rounded version of the exact sqrt(5), but disagrees if both are rounded to 5 decimal digits, where 2.23607 <> 2.23611.
So a(3) = 4 (digits), the maximum rounding level of agreement.

Cf. A138335, A138336, A138337, A138339, A138343, A138366, A138369, A138370.

Definition and values replaced as defined via continued fractions by R. J. Mathar, Oct 01 2009

A379136 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a pentakis dodecahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 17 2024

The pentakis dodecahedron is the dual polyhedron of the truncated icosahedron.

			2.7352547614903346619898560183934957927169693...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pentakis Dodecahedron.
Wikipedia, Pentakis dodecahedron.

Cf. A379132 (surface area), A379133 (volume), A379134 (inradius), A379135 (midradius).
Cf. A236367 and A344075 (dihedral angles of a truncated icosahedron).
Cf. A002163.

First[RealDigits[ArcCos[-(80 + 9*Sqrt[5])/109], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["PentakisDodecahedron", "DihedralAngles"]], 10, 100]]

acos(-(80 + 9*sqrt(5))/109) \\ Charles R Greathouse IV, Feb 05 2025

Equals arccos(-(80 + 9*sqrt(5))/109) = arccos(-(80 + 9*A002163)/109).

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A378976 Decimal expansion of the midradius of a triakis icosahedron with unit shorter edge length.

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A378977 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a triakis icosahedron.

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A379132 Decimal expansion of the surface area of a pentakis dodecahedron with unit shorter edge length.

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A379133 Decimal expansion of the volume of a pentakis dodecahedron with unit shorter edge length.

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A379387 Decimal expansion of the inradius of a deltoidal hexecontahedron with unit shorter edge length.

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A379389 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a deltoidal hexecontahedron.

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A384138 Decimal expansion of the volume of an elongated pentagonal pyramid with unit edge.

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