A002163 -id:A002163 - cp's OEIS frontend

A378974 Decimal expansion of the volume of a triakis icosahedron with unit shorter edge length.

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

Offset: 2

Paolo Xausa, Dec 14 2024

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

			12.017220926874316513329814423376647765182009668737...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Triakis Icosahedron.
Wikipedia, Triakis icosahedron.

Cf. A378973 (surface area), A378975 (inradius), A378976 (midradius), A378977 (dihedral angle).
Cf. A377695 (volume of a truncated dodecahedron with unit edge length).
Cf. A002163.

First[RealDigits[(19 + 13*Sqrt[5])/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TriakisIcosahedron", "Volume"], 10, 100]]

Equals (19 + 13*sqrt(5))/4 = (19 + 13*A002163)/4.

A379385 Decimal expansion of the surface area of a deltoidal hexecontahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Dec 22 2024

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

			92.231912906404640710406169319098384407207052545184...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Deltoidal Hexecontahedron.
Wikipedia, Deltoidal hexecontahedron.

Cf. A379386 (volume), A379387 (inradius), A379388 (midradius), A379389 (dihedral angle).
Cf. A344149 (surface area of a (small) rhombicosidodecahedron with unit edge length).
Cf. A002163.

First[RealDigits[Sqrt[4370 + 1850*Sqrt[5]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["DeltoidalHexecontahedron", "SurfaceArea"], 10, 100]]

Equals sqrt(4370 + 1850*sqrt(5)) = sqrt(4370 + 1850*A002163).

A379386 Decimal expansion of the volume of a deltoidal hexecontahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Dec 23 2024

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

			81.004143635377089099456665341616282246804393456803...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Deltoidal Hexecontahedron.
Wikipedia, Deltoidal hexecontahedron.

Cf. A379385 (surface area), A379387 (inradius), A379388 (midradius), A379389 (dihedral angle).
Cf. A185093 (volume of a (small) rhombicosidodecahedron with unit edge length).
Cf. A002163.

First[RealDigits[Sqrt[(29530 + 13204*Sqrt[5])/9], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["DeltoidalHexecontahedron","Volume"], 10, 100]]

Equals sqrt((29530 + 13204*sqrt(5))/9) = sqrt((29530 + 13204*A002163)/9).

A379710 Decimal expansion of the inradius of a disdyakis triacontahedron with unit shorter edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 31 2024

The disdyakis triacontahedron is the dual polyhedron of the truncated icosidodecahedron (great rhombicosidodecahedron).

			2.679969340204835578579553327459806767085423038168...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Disdyakis Triacontahedron.
Wikipedia, Disdyakis triacontahedron.

Cf. A379708 (surface area), A379709 (volume), A379388 (midradius), A379711 (dihedral angle).

Cf. A002163.

First[RealDigits[Sqrt[3477/964 + 7707/(964*Sqrt[5])], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["DisdyakisTriacontahedron", "Inradius"], 10, 100]]

Equals sqrt(3477/964 + 7707/(964*sqrt(5))) = sqrt(3477/964 + 7707/(964*A002163)).

A379711 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a disdyakis triacontahedron.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 31 2024

The disdyakis triacontahedron is the dual polyhedron of the truncated icosidodecahedron (great rhombicosidodecahedron).

			2.8778366104612242809434504548179917754749428665406...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Disdyakis Triacontahedron.
Wikipedia, Disdyakis triacontahedron.

Cf. A379708 (surface area), A379709 (volume), A379710 (inradius), A379388 (midradius).
Cf. A344075, A377995 and A377996 (dihedral angles of a truncated icosidodecahedron (great rhombicosidodecahedron)).
Cf. A002163.

First[RealDigits[ArcCos[(-179 - 24*Sqrt[5])/241], 10, 100]] (* or *)
First[RealDigits[First[PolyhedronData["DisdyakisTriacontahedron", "DihedralAngles"]], 10, 100]]

Equals arccos((-179 - 24*sqrt(5))/241) = arccos((-179 - 24*A002163)/241).

A384284 Decimal expansion of the surface area of a gyroelongated pentagonal cupola with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, May 27 2025

The gyroelongated pentagonal cupola is Johnson solid J_24.

			25.240003790832583513731278051892586452816662365...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Gyroelongated pentagonal cupola.
Index entries for algebraic numbers, degree 8

Cf. A384283 (volume).

Cf. A002163, A002194, A179553, A179591, A179640, A384141.

First[RealDigits[(20 + 25*Sqrt[3] + Sqrt[725 + 310*Sqrt[5]])/4, 10, 100]]
First[RealDigits[PolyhedronData["J24", "SurfaceArea"], 10, 100]]

(20 + 25*sqrt(3) + sqrt(725 + 310*sqrt(5)))/4 \\ Charles R Greathouse IV, Aug 19 2025

Equals (20 + 25*sqrt(3) + sqrt(725 + 310*sqrt(5)))/4 = (20 + 25*A002194 + sqrt(725 + 310*A002163))/4.

Equals the largest root of 256*x^8 - 10240*x^7 + 12800*x^6 + 3200000*x^5 - 22476000*x^4 - 203280000*x^3 + 1412362500*x^2 + 3080375000*x - 17984046875.

A134945 Decimal expansion of 1 + sqrt(5).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 15 2007

If "index" equals (0,2) then this sequence is the decimal expansion of (golden ratio divided by 5 = phi/5 = (1 + sqrt(5))/10). Example: 0.323606797...
Apart from the leading digit the same as A134972, A098317 and A002163. - R. J. Mathar, Aug 06 2013
Length of the longest diagonal in a regular 10-gon with unit side. - Mohammed Yaseen, Nov 12 2020
Abscissa of the first superstable point of the logistic map (see Finch). - Stefano Spezia, Nov 23 2024

			3.2360679774997896964...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.9, p. 66.

Index entries for algebraic numbers, degree 2.

Cf. A000045, A002163, A098317, A134972.

RealDigits[1 + Sqrt[5], 10, 100][[1]] (* Michael De Vlieger, Nov 13 2020 *)

PARI
```
1 + sqrt(5) \\ Altug Alkan, Mar 19 2018
```

From Christian Katzmann, Mar 19 2018: (Start)
Equals Sum_{n>=0} (15*(2*n)!+8*n!^2)/(n!^2*3^(2*n+2)).
Equals 1 + Sum_{n>=0} 5*(2*n)!/(n!^2*3^(2*n+1)). (End)
Equals 1/A019827. - R. J. Mathar, Jan 17 2021
Equals Product_{k>=1} (1 + 1/Fibonacci(2*k)). - Amiram Eldar, May 27 2021

More terms from Jinyuan Wang, Mar 30 2020

A176015 Decimal expansion of (5 + 3*sqrt(5))/10.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 06 2010

Continued fraction expansion of (5 + 3*sqrt(5))/10 is A010686.

The horizontal distance between the accumulation point and the outermost point of a golden spiral inscribed inside a golden rectangle with dimensions phi and 1 along the x and y axes, respectively (the vertical distance is A244847). - Amiram Eldar, May 18 2021

			(5 + 3*sqrt(5))/10 = 1.17082039324993690892...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.2 The Golden Mean, phi, p. 7.

Muniru A Asiru, Table of n, a(n) for n = 1..1000 [a(1000) corrected by _Georg Fischer_, Apr 02 2020]
Index entries for algebraic numbers, degree 2.

Cf. A000032, A000045, A001622, A002163 (decimal expansion of sqrt(5)), A010686 (repeat 1, 5), A090550, A134976.

Cf. A010499 (decimal expansion of 3*sqrt(5)).

SetDefaultRealField(RealField(105)); n:=(5+3*Sqrt(5))/10; Reverse(Intseq(Floor(10^104*n))); // Arkadiusz Wesolowski, Jan 07 2018

Digits := 1000:  (5+3*sqrt(5.0))/10; # Muniru A Asiru, Jan 22 2018

RealDigits[(5 + 3 Sqrt[5])/10, 10, 1001][[1]] (* Georg Fischer, Apr 02 2020 *)

(5 + 3*sqrt(5))/10 \\ Michel Marcus, Apr 20 2020

Equals (A134976 + 8)/10. - R. J. Mathar, Apr 12 2010
From Arkadiusz Wesolowski, Jan 07 2018: (Start)
Equals A001622^2 / sqrt(5).
Equals lim_{n -> infinity} A000045(n+2) / A001622^n. (End)
Equals 1/A090550 + 1. - Michel Marcus, Apr 20 2020
Minimal polynomial is 5x^2 - 5x - 1 (this number is an algebraic number but not an algebraic integer). - Alonso del Arte, Apr 20 2020
Equals lim_{k->oo} Fibonacci(k+2)/Lucas(k). - Amiram Eldar, Feb 06 2022

A176055 Decimal expansion of (2+sqrt(5))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 07 2010

Continued fraction expansion of (2+sqrt(5))/2 is A010698.

a(n) = A020837(n-1) for n > 1; a(1) = 2.

			2.11803398874989484820...

Index entries for algebraic numbers, degree 2.

Cf. A002163 (sqrt(5)), A020837 (1/sqrt(80)), A010698 (repeat 2, 8).

Cf. A000045, A001622.

RealDigits[GoldenRatio + 1/2, 10, 100][[1]] (* Amiram Eldar, Jun 06 2021 *)

Equals 1/2 + phi, with phi = A001622.
From Amiram Eldar, Jun 06 2021: (Start)
Equals 1 + Sum{k>=0} 1/(Fibonacci(2*k+1)+1).
Equals 1 + Sum{k>=0} binomial(2*k,k)/20^k. (End)

A377695 Decimal expansion of the volume of a truncated dodecahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 04 2024

			85.039664559370881554679651012654159610712109542...

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

Cf. A377694 (surface area), A377696 (circumradius), A377697 (midradius), A377698 (Dehn invariant, negated).
Cf. A102769 (analogous for a regular dodecahedron).
Cf. A002163.

First[RealDigits[5/12*(99 + 47*Sqrt[5]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedDodecahedron", "Volume"], 10, 100]]

Equals (5/12)*(99 + 47*sqrt(5)) = (5/12)*(99 + 47*A002163).

cp's OEIS Frontend

A378974 Decimal expansion of the volume of a triakis icosahedron with unit shorter edge length.

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A379385 Decimal expansion of the surface area of a deltoidal hexecontahedron with unit shorter edge length.

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A379386 Decimal expansion of the volume of a deltoidal hexecontahedron with unit shorter edge length.

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A379710 Decimal expansion of the inradius of a disdyakis triacontahedron with unit shorter edge length.

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A379711 Decimal expansion of the dihedral angle, in radians, between any two adjacent faces in a disdyakis triacontahedron.

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A384284 Decimal expansion of the surface area of a gyroelongated pentagonal cupola with unit edge.

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A134945 Decimal expansion of 1 + sqrt(5).

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