A010469 -id:A010469 - cp's OEIS frontend

A384287 Decimal expansion of the volume of a square orthobicupola with unit edge.

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

Offset: 1

Paolo Xausa, Jun 05 2025

The square orthobicupola is Johnson solid J_28.

Also the volume of a square gyrobicupola (Johnson solid J_29) with unit edge.

			3.885618083164126731735584965612930771426229167169...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Square gyrobicupola.
Wikipedia, Square orthobicupola.

Cf. A010469 (surface area - 10).

Cf. A002193, A010487, A384624.

First[RealDigits[2 + Sqrt[32]/3, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J28", "Volume"], 10, 100]]

Equals 2 + (4/3)*sqrt(2) = 2 + (4/3)*A002193 = 2 + A010487/3.

Equals the largest root of 9*x^2 - 36*x + 4.

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.

A131266 Decimal expansion of 2sqrt(3)log(2)/Pi.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Sep 28 2007

Also: a constant describing the peak location of the density of states of the minimal difference partition problem in the fermionic case [Comtet et al.].

			0.76430413884568819720562499904060001690455623711504906130392...

Gheorghe Coserea, Table of n, a(n) for n = 0..12345
A. Comtet, S. N. Majumdar and S. Ouvry, Integer Partitions and Exclusion Statistics, arXiv:0705.2640 [cond-mat.stat-mech], 2007, eq (6).

Cf. A030699, A257639.

RealDigits[Sqrt[3] Log[4]/Pi, 10, 111][[1]] (* Robert G. Wilson v, Nov 08 2015 *)

PARI
```
print(2*sqrt(3)*log(2)/Pi);
```

default(realprecision, 60);
eval(vecextract(Vec(Str(2*sqrt(3)*log(2)/Pi)), "3..-2"))  \\ Gheorghe Coserea, Nov 07 2015

Equals A010469*A002162/A000796.

Equals lim A257639(n)/sqrt(n) when n tends to infinity.

Leading zero removed by R. J. Mathar, Feb 06 2009

A386461 Decimal expansion of the surface area of a biaugmented truncated cube with unit edges.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jul 23 2025

The biaugmented truncated cube is Johnson solid J_67.

			36.241911729260269564523295159701074096328596018257...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Biaugmented truncated cube.

Cf. A010524 (volume - 9).

Cf. A010469, A010487, A010502, A377342, A385535, A386460.

First[RealDigits[18 + 8*Sqrt[2] + Sqrt[48], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J67", "SurfaceArea"], 10, 100]]

Equals 2*(9 + 4*sqrt(2) + 2*sqrt(3)) = 2*(9 + A010487 + A010469) = 18 + A377342 + A010502.

Equals the largest root of x^4 - 72*x^3 + 1592*x^2 - 10656*x - 2672.

A017940 Powers of sqrt(12) rounded down.

Original entry on oeis.org

1, 3, 12, 41, 144, 498, 1728, 5985, 20736, 71831, 248832, 861979, 2985984, 10343751, 35831808, 124125023, 429981696, 1489500287, 5159780352, 17874003451, 61917364224, 214488041413, 743008370688

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A010469 (sqrt(12)).

[Floor(Sqrt(12^n)): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

Floor[(Sqrt[12])^Range[0,30]] (* Harvey P. Dale, Aug 26 2019 *)

a(n)=sqrtint(12^n) \\ Charles R Greathouse IV, Nov 18 2011

a(n) = floor(sqrt(12^n)). - Vincenzo Librandi, Jun 24 2011

A171541 Decimal expansion of 2*sqrt(3/35).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 11 2009

The absolute value of the Clebsch-Gordan coupling coefficient = <2 3/2 ; -1 1/2 | 7/2 -1/2>.

			sqrt(12/35) = 2*sqrt(105)/35 = 0.5855400437691199076126307817440601137562...

Daniel Starodubtsev, Table of n, a(n) for n = 0..10000
Wikipedia, Table of Clebsch-Gordan coefficients

Cf. A010469, A010490, A171546.

RealDigits[2*Sqrt[3/35],10,120][[1]] (* Harvey P. Dale, Feb 29 2012 *)

Equals 2*A171546 = A010469/A010490.

A381687 Decimal expansion of the isoperimetric quotient of a truncated octahedron.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Mar 04 2025

For the definition of isoperimetric quotient of a solid, references and links, see A381684.

			0.753366625166156882229489414578751361927704595866...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

Cf. A377341 (surface area), A377342 (volume).

Cf. A000796, A010469, A381684.

First[RealDigits[64*Pi/(3*(1 + 2*Sqrt[3])^3), 10, 100]]

Equals 36*Pi*A377342^2/(A377341^3).

Equals 64*Pi/(3*(1 + 2*sqrt(3))^3) = 64*A000796/(3*(1 + A010469)^3).

A386753 Decimal expansion of the surface area of a bilunabirotunda with unit edges.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Aug 01 2025

The bilunabirotunda is Johnson solid J_91.

			12.346011217493622278090940592566183131515107...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Bilunabirotunda.

Cf. A189963 (volume - 1).

Cf. A010469, A010476.

First[RealDigits[2 + Sqrt[12] + Sqrt[5*(5 + Sqrt[20])], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J91", "SurfaceArea"], 10, 100]]

Equals 2 + 2*sqrt(3) + sqrt(5*(5 + 2*sqrt(5))) = 2 + A010469 + sqrt(5*(5 + A010476)).

Equals the largest root of x^8 - 16*x^7 - 36*x^6 + 1328*x^5 - 2946*x^4 - 16624*x^3 + 34796*x^2 + 61392*x - 10679.

A140457 Decimal expansion of surface area of unit elongated dodecahedron.

Original entry on oeis.org

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

Offset: 2

Jonathan Vos Post, Jun 26 2008

			18.138271537828097532....

Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, 1973.
Fedorov, E. S. "Elemente der Gestaltenlehre." Mineralogicheskoe obshchestvo Leningrad (Verhandlungen der Russisch-Kaiserlichen Mineralogischen Gesellschaft zu St. Petersburg 21, 1-279, 1885.
Tutton, A. E. H. Crystallography and Practical Crystal Measurement. London, pp. 567 (Fig. 448) and 723 (Fig. 585), 1922.

Eric W. Weisstein, Elongated Dodecahedron.

RealDigits[2*Sqrt[3]*(3+Sqrt[5]),10,130][[1]] (* Harvey P. Dale, Nov 26 2011 *)

2*sqrt(3)*(3 + sqrt(5)) = A010469*(1+A098317).

Corrected offset. Added more digits R. J. Mathar, Jan 26 2009

cp's OEIS Frontend

A384287 Decimal expansion of the volume of a square orthobicupola with unit edge.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A131266 Decimal expansion of 2*sqrt(3)*log(2)/Pi.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A386461 Decimal expansion of the surface area of a biaugmented truncated cube with unit edges.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A017940 Powers of sqrt(12) rounded down.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A171541 Decimal expansion of 2*sqrt(3/35).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A381687 Decimal expansion of the isoperimetric quotient of a truncated octahedron.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A131266 Decimal expansion of 2sqrt(3)log(2)/Pi.