A002194 -id:A002194 - cp's OEIS frontend

A165663 Decimal expansion of 3 + sqrt(3).

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

Offset: 1

Jonathan Vos Post, Sep 24 2009

Arises as an upper limit of indices of subfactors in the extended Haagerup planar algebra (see Bigelow et al.)
Perimeter of a 30-60-90 triangle with shortest side equal to 1. - Wesley Ivan Hurt, Apr 09 2016
Surface area of an elongated triangular pyramid (Johnson solid J_7) with unit edges. - Paolo Xausa, Aug 02 2025

			4.732050807568877293527446341505872366942805253810380628...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Stephen Bigelow, Scott Morrison, Emily Peters, and Noah Snyder, Constructing the extended Haagerup planar algebra, arXiv:0909.4099 [math.OA], 2009-2011.
Wikipedia, Elongated triangular pyramid.
Index entries for algebraic numbers, degree 2

Cf. A002194, A019973, A090388, A160390.

SetDefaultRealField(RealField(100)); 3 + Sqrt(3); // G. C. Greubel, Nov 20 2018

Digits:=100: evalf(3+sqrt(3)); # Wesley Ivan Hurt, Apr 09 2016

RealDigits[3 + Sqrt[3], 10, 100][[1]] (* Wesley Ivan Hurt, Apr 09 2016 *)

default(realprecision, 100); 3 + sqrt(3) \\ G. C. Greubel, Nov 20 2018

numerical_approx(3+sqrt(3), digits=100) # G. C. Greubel, Nov 20 2018

Equals 4 + A160390 = 1 + A019973 = 2 + A090388 = 3 + A002194. - R. J. Mathar, Sep 27 2009

A232812 Decimal expansion of the surface index of a regular tetrahedron.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Dec 01 2013

Equivalently, the surface area of a regular tetrahedron with unit volume. Among Platonic solids, surface indices decrease with increasing number of faces: this one, 6.0 (cube = hexahedron), A232811 (octahedron), A232810 (dodecahedron), and A232809 (icosahedron).

			7.20562173105601636005279232409725707779044450935589335...

Stanislav Sykora, Table of n, a(n) for n = 1..1000
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 6

Cf. A002194, A020829, A232808 (surface index of a sphere), A232809, A232810, A232811.

RealDigits[2*Sqrt[3]*3^(2/3), 10, 120][[1]] (* Amiram Eldar, May 25 2023 *)

sqrtn(139968,6) \\ Charles R Greathouse IV, Apr 25 2016

Equals 2*sqrt(3)*3^(2/3).

Equals A002194/A020829^(2/3).

A234519 Natural numbers n sorted by decreasing values of number k(n) = sigma(n)^(1/n), where sigma(n) = A000203(n) = the sum of divisors of n.

Original entry on oeis.org

2, 4, 3, 6, 5, 8, 7, 10, 9, 12, 14, 11, 16, 15, 18, 13, 20, 24, 17, 21, 22, 19, 28, 26, 30, 23, 25, 27, 32, 36, 34, 33, 29, 40, 31, 35, 42, 38, 39, 44, 48, 37, 45, 46, 41, 50, 54, 52, 43, 56, 60, 51, 49, 47, 55, 58, 57, 64, 66, 53, 63, 62, 72, 68, 70, 59, 65

Offset: 1

Jaroslav Krizek, Jan 04 2014

nonn

Number k(n) = sigma(n)^(1/n) is number such that k(n)^n = sigma(n).
For number 2; k(2) = sigma(2)^(1/2) = sqrt(3) = 1,732050807568… = A002194 (maximal value of function k(n)).
The last term of this infinite sequence is number 1, k(1) = 1 (minimal value of function k(n)).
Conjecture: Every natural number n has a unique value of number k(n).
See A234521 - sequence of numbers a(n) such that a(n) > a(k) for all k < n.

Jaroslav Krizek, Table of n, a(n) for n = 1..1000

Cf. A234515, A234516, A234517, A234518, A234520, A234521, A234522, A234523, A234524.

a(n)=vecsort(vector(2*n, i, sigma(i)^(1/i)), ,5)[n] \\ Michel Marcus and Ralf Stephan, Jan 14 2014

A385257 Decimal expansion of the surface area of a gyroelongated triangular bicupola with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jun 24 2025

The gyroelongated triangular bicupola is Johnson solid J_44.

			14.660254037844386467637231707529361834714026269...

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

Cf. A385256 (volume).
Cf. A002194, A010527.
Essentially the same of A332133, A375193 and A010527.

First[RealDigits[6 + 5*Sqrt[3], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J44", "SurfaceArea"], 10, 100]]

Equals 6 + 5*sqrt(3) = 6 + 5*A002194 = 6 + 10*A010527.

Equals the largest root of x^2 - 12*x - 39.

A118292 Decimal expansion of (Gamma(1/6)Gamma(1/3))/(3sqrt(Pi)).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Apr 22 2006

General formula: Integral_{x=0..1} (1+x^(3n))/sqrt(1-x^3) dx = G_3 * k_n = G_3*A146751(n)/A146752(n) = A118292*A146751(n)/A146752(n) where G_3 = (Gamma(1/3)^3)/(2^(1/3)*sqrt(3)*Pi) is the number in the present entry. For numerators of k_n see A146752, for denominators of k_n see A146753. - Artur Jasinski

gamma(1/6)*gamma(1/3)/(3*sqrt(Pi)) = gamma(1/3)^3/(2^(1/3)*sqrt(3)*Pi). - Harry J. Smith, May 09 2009

			2.8043642106509085223500381581005882709260444108....

Harry J. Smith, Table of n, a(n) for n = 1..4000
Eric Weisstein's World of Mathematics, Butterfly Curve

Cf. A146752, A146753, A160323 (continued fraction).

RealDigits[(Gamma[1/3]^3)/(2^(1/3) Sqrt[3] Pi), 10, 200] (* Artur Jasinski*)

allocatemem(932245000); default(realprecision, 4080); x=gamma(1/3)^3/(2^(1/3)*sqrt(3)*Pi); for (n=1, 4000, d=floor(x); x=(x-d)*10; write("b118292.txt", n, " ", d)) \\ Harry J. Smith, Jun 20 2009

3/hypergeom([1/3,1/6],[3/2],1) \\ Charles R Greathouse IV, Aug 29 2025

Equals A073005^3 / (A002194*A002580*A000796) [see Vidunas, arXiv:math.CA/0403510]. - R. J. Mathar, Nov 30 2008

Equals 3/hypergeom([1/3, 1/6], [3/2], 1) = A290570*A005480. - Peter Bala, Mar 02 2022

Edited by N. J. A. Sloane, Nov 16 2008 at the suggestion of R. J. Mathar

A187110 Decimal expansion of sqrt(3/8).

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 05 2011

Apart from leading digits, the same as A174925.

Radius of the circumscribed sphere (congruent with vertices) for a regular tetrahedron with unit edges. - Stanislav Sykora, Nov 20 2013

			sqrt(3/8)=0.61237243569579452454932101867647284799148687016417..

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Michael Penn, Maximize the area of one ellipse!, YouTube video, 2020.
Wikipedia, Tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. A002194, A010464, A010466, A020781, A040001, A040005, A115754, A301755.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A019881 (icosahedron), A179296 (dodecahedron). - Stanislav Sykora, Feb 10 2014

Mathematica
```
RealDigits[N[Sqrt[3/8],300]][[1]]
```

sqrt(3/8) \\ Charles R Greathouse IV, Mar 16 2022

Equals A010464/4. - Stefano Spezia, Jan 26 2025

Equals 3*A020781 = A115754/2 = sqrt(A301755). - Hugo Pfoertner, Jan 26 2025

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.

A385259 Decimal expansion of the surface area of a gyroelongated square bicupola with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Jun 26 2025

The gyroelongated square bicupola is Johnson solid J_45.

			20.392304845413263761164678049035234201656831522862...

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

Cf. A385258 (volume).

Cf. A002194, A179588, A384215, A385257.

First[RealDigits[10 + 6*Sqrt[3], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J45", "SurfaceArea"], 10, 100]]

Equals 10 + 6*sqrt(3) = 10 + 6*A002194.

Equals the largest root of x^2 - 20*x - 8.

A171972 Greatest integer k such that k/n^2 < sqrt(3).

Original entry on oeis.org

0, 1, 6, 15, 27, 43, 62, 84, 110, 140, 173, 209, 249, 292, 339, 389, 443, 500, 561, 625, 692, 763, 838, 916, 997, 1082, 1170, 1262, 1357, 1456, 1558, 1664, 1773, 1886, 2002, 2121, 2244, 2371, 2501, 2634, 2771, 2911, 3055, 3202, 3353, 3507, 3665, 3826, 3990, 4158

Offset: 0

Reinhard Zumkeller, Jan 20 2010

nonn

Integer part of the surface area of a regular tetrahedron with edge length n.
A171970(n)*A005843(n) <= a(n);
a(n) <= 4*A171971(n); 0 <= a(n) - 4*A171971(n) < 4.

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Tetrahedron
Wikipedia, Tetrahedron

Cf. A002194, A070169, A171974, A171973, A171975, A000290, A293410.

a171972 = floor . (* sqrt 3) . fromInteger . a000290
-- Reinhard Zumkeller, Dec 15 2012

z = 120; r = Sqrt[3];
Table[Floor[r*n^2], {n, 0, z}]; (* A171972 *)
Table[Ceiling[r*n^2], {n, 0, z}]; (* A293410 *)
Table[Round[r*n^2], {n, 0, z}]; (* A070169. -  Clark Kimberling, Oct 11 2017 *)

a(n) = floor(n^2 * sqrt(3)).
a(n) = A022838(n^2);
a(n) = A293410(n) - 1 for n > 0.

A377694 Decimal expansion of the surface area of a truncated dodecahedron with unit edge length.

Original entry on oeis.org

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

Offset: 3

Paolo Xausa, Nov 04 2024

			100.990760153101988544745948988636656554915090575...

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

Cf. A377695 (volume), A377696 (circumradius), A377697 (midradius), A377698 (Dehn invariant, negated).
Cf. A131595 (analogous for a regular dodecahedron).
Cf. A002194, A010476.

First[RealDigits[5*(Sqrt[3] + 6*Sqrt[5 + Sqrt[20]]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedDodecahedron", "SurfaceArea"], 10, 100]]

Equals 5*(sqrt(3) + 6*sqrt(5 + 2*sqrt(5))) = 5*(A002194 + 6*sqrt(5 + A010476)).

cp's OEIS Frontend

A165663 Decimal expansion of 3 + sqrt(3).

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A232812 Decimal expansion of the surface index of a regular tetrahedron.

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A234519 Natural numbers n sorted by decreasing values of number k(n) = sigma(n)^(1/n), where sigma(n) = A000203(n) = the sum of divisors of n.

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A385257 Decimal expansion of the surface area of a gyroelongated triangular bicupola with unit edge.

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A118292 Decimal expansion of (Gamma(1/6)*Gamma(1/3))/(3*sqrt(Pi)).

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A187110 Decimal expansion of sqrt(3/8).

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

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A118292 Decimal expansion of (Gamma(1/6)Gamma(1/3))/(3sqrt(Pi)).