A002194 -id:A002194 - cp's OEIS frontend

A293327 Greatest integer k such that k/2^n < sqrt(1/3).

0, 1, 2, 4, 9, 18, 36, 73, 147, 295, 591, 1182, 2364, 4729, 9459, 18918, 37837, 75674, 151348, 302697, 605395, 1210791, 2421582, 4843165, 9686330, 19372660, 38745320, 77490641, 154981282, 309962565, 619925131, 1239850262, 2479700524, 4959401049, 9918802098

Offset: 0

Clark Kimberling, Oct 09 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A002194, A094386, A293328, A293329.

z = 120; r = Sqrt[1/3];
Table[Floor[r*2^n], {n, 0, z}];   (* A293327 *)
Table[Ceiling[r*2^n], {n, 0, z}]; (* A293328 *)
Table[Round[r*2^n], {n, 0, z}];   (* A293329 *)

a(n) = floor(r*2^n), where r = sqrt(1/3).

a(n) = A293328(n) - 1.

Definition and formula corrected by Clark Kimberling, Dec 26 2022

A293329 The integer k that minimizes |k/2^n - sqrt(1/3)|.

Original entry on oeis.org

1, 1, 2, 5, 9, 18, 37, 74, 148, 296, 591, 1182, 2365, 4730, 9459, 18919, 37837, 75674, 151349, 302698, 605396, 1210791, 2421583, 4843165, 9686330, 19372660, 38745321, 77490641, 154981283, 309962566, 619925131, 1239850262, 2479700525, 4959401049, 9918802098

Offset: 0

Clark Kimberling, Oct 10 2017

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A002194, A094386, A293327, A293328.

z = 120; r = Sqrt[1/3];
Table[Floor[r*2^n], {n, 0, z}];   (* A293327 *)
Table[Ceiling[r*2^n], {n, 0, z}]; (* A293328 *)
Table[Round[r*2^n], {n, 0, z}]; (* A293329 *)

a(n) = floor(1/2 + r*2^n), where r = sqrt(1/3).

a(n) = A293327(n) if (fractional part of r*2^n) < 1/2, else a(n) = A293328(n).

A354129 Decimal expansion of 7 + 4*sqrt(3).

Original entry on oeis.org

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

Offset: 2

Stefano Spezia, May 18 2022

The largest root of x^2 - 14*x + 1 = 0.

Apart from leading digits the same as A010502. - R. J. Mathar, May 24 2022

			13.92820323027550917410978536602...

Nicholas Owad and Anastasiia Tsvietkova, Random meander model for links, arXiv:2205.03451 [math.GT], 2022, p. 13.

Cf. A002194, A010502, A019973 (square root), A354128 (multiplicative inverse).

Mathematica
```
First[RealDigits[N[7+4Sqrt[3],92]]]
```

Equals (1 + sqrt(3))^4 / 4. - Vaclav Kotesovec, May 18 2022
Equals (2 + sqrt(3))^2 = A019973^2. - Jianing Song, May 27 2022
Equals exp(arccosh(7)). - Amiram Eldar, Jul 06 2023

A374948 Decimal expansion of the Euclidean length of the minimum Steiner tree joining all the vertices of a unit cube.

Original entry on oeis.org

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

Offset: 1

Marco Ripà, Jul 24 2024

The 1994 Bridge's paper entitled "Minimal Steiner Trees for Three Dimensional Networks" (see Links) suggested an optimal strategy to solve the minimum Steiner tree problem for the unit cube {0,1}^3, and the total length of the provided Steiner Tree is 1 + 3*sqrt(3).

Also the surface area of a gyroelongated square pyramid (Johnson solid J_10) with unit edges. - Paolo Xausa, Aug 04 2025

			6.1961524227066318805823390245176171008284157614311418841674209383...

R. Bridges, Minimal Steiner Trees for Three Dimensional Networks, Math. Gaz., 78 (1994), 157-162.
Math Overflow, Joining the 2^k points of {0,1}^k with the shortest tree.
Mathematics Stack Exchange, Steiner tree problem in 3D.
J. M. Smith, R. Weiss, and M. Patel, An O(N2) Heuristic for Steiner Minimal Trees in E3, Networks 26 (1995), 273-289.
B. Toppur and J. M. A. Smith, A Sausage Heuristic for Steiner Minimal Trees in Three-Dimensional Euclidean Space, J. Math. Modelling and Algorithms, 4 (2005), 199-217.
Wikipedia, Gyroelongated square pyramid.

Cf. A002194, A010482, A374148, A374260.

Essentially the same as A178809, A176532 and A010482.

RealDigits[3Sqrt[3]+1,10,87][[1]] (* Stefano Spezia, Jul 25 2024 *)

Equals 3*sqrt(3) + 1.

Equals A010482(n) for any n >= 2 and a(1) = A010482(1) + 1.

A376859 Decimal expansion of Product_{k=1..4} Gamma(k/3).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Oct 09 2024

			3.23937134071697320618006601163079489801213782...

Eric Weisstein's World of Mathematics, Gamma Function.
Index to sequences related to the Gamma function.

Cf. A002194, A019692, A202623.

Other identities for Product_{k=1..m} Gamma(k/3): A073005 (m = 1), A186706 (m = 2 and m = 3), A376911 (m = 5 and m = 6), A376912 (m = 7), A376913 (m = 8).

First[RealDigits[2*Pi*Gamma[4/3]/Sqrt[3], 10, 100]]

Equals 2*Pi*Gamma(1/3)/(3*sqrt(3)) = 2*Pi*Gamma(4/3)/sqrt(3) = A186706*A202623 (cf. eq. 86 in Weisstein link).

A376913 Decimal expansion of Product_{k=1..8} Gamma(k/3).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Oct 11 2024

			5.2386596251856584103292320999763662681359773992...

Eric Weisstein's World of Mathematics, Gamma Function.
Index to sequences related to the Gamma function.

Cf. A002194, A091925.

Other identities for Product_{k=1..m} Gamma(k/3): A073005 (m = 1), A186706 (m = 2 and m = 3), A376859 (m = 4), A376911 (m = 5 and m = 6), A376912 (m = 7).

First[RealDigits[640*Pi^3/(2187*Sqrt[3]), 10, 100]]

Equals 640*Pi^3/(2187*sqrt(3)) = 640*A091925/(3^7*A002194) (cf. eq. 90 in Weisstein link).

A377298 Decimal expansion of the surface area of a truncated cube with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 25 2024

			32.4346643636148951726751573735281216767216730121...

Eric Weisstein's World of Mathematics, Truncated Cube.
Wikipedia, Truncated cube.

Cf. A377299 (volume), A294968 (circumradius), A010503 (midradius - 1), A377296 (Dehn invariant, negated).

Cf. A002193, A002194, A010469, A010524.

First[RealDigits[2*(6 + Sqrt[72] + Sqrt[3]), 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedCube", "SurfaceArea"], 10, 100]]

Equals 2*(6 + 6*sqrt(2) + sqrt(3)) = 2*(6 + 2*A002193 + A002194) = 12 + 2*A010524 + A010469.

A377341 Decimal expansion of the surface area of a truncated octahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Oct 25 2024

			26.78460969082652752232935609807046840331366304572...

Eric Weisstein's World of Mathematics, Truncated Octahedron.
Wikipedia, Truncated octahedron.

Cf. A377342 (volume), A020797 (circumradius/10), A152623 (midradius).
Cf. A010469 (analogous for a regular octahedron).
Cf. A002194.

First[RealDigits[6 + 12*Sqrt[3], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedOctahedron", "SurfaceArea"], 10, 100]]

Equals 6 + 12*sqrt(3) = 6 + 12*A002194.

A377557 Decimal expansion of 2Pi^3/(81sqrt(3)) + 13*zeta(3)/27.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 01 2024

			1.0207800444333631028232547399039818253534109375...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 52.

Cf. A002117, A002194, A016777, A016779, A091925, A233091, A377558, A377559, A377560.

RealDigits[2Pi^3/(81Sqrt[3])+13Zeta[3]/27,10,100][[1]]

Equals Sum_{k>=0} 1/(3*k + 1)^3 (see Finch).
Equals -psi''(1/3)/54 (see Shamos).
Equals hypergeom([1/3, 1/3, 1/3, 1], [4/3, 4/3, 4/3], 1). - R. J. Mathar, Jul 14 2025

A377560 Decimal expansion of Pi^3/(36sqrt(3)) + 91zeta(3)/216.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 01 2024

			1.00368551534795269706323013702486057315272784359...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 28.

Cf. A002117, A002194, A016921, A016923, A091925, A233091, A377557, A377558, A377559.

RealDigits[Pi^3/(36*Sqrt[3])+91*Zeta[3]/216,10,100][[1]]

Equals Sum_{k>=0} 1/(6*k + 1)^3 (see Finch).

Equals -psi''(1/6)/432 (see Shamos).

cp's OEIS Frontend

A293327 Greatest integer k such that k/2^n < sqrt(1/3).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A293329 The integer k that minimizes |k/2^n - sqrt(1/3)|.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A354129 Decimal expansion of 7 + 4*sqrt(3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A374948 Decimal expansion of the Euclidean length of the minimum Steiner tree joining all the vertices of a unit cube.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A376859 Decimal expansion of Product_{k=1..4} Gamma(k/3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A376913 Decimal expansion of Product_{k=1..8} Gamma(k/3).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A377298 Decimal expansion of the surface area of a truncated cube with unit edge length.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A377341 Decimal expansion of the surface area of a truncated octahedron with unit edge length.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

A377557 Decimal expansion of 2Pi^3/(81sqrt(3)) + 13*zeta(3)/27.

A377560 Decimal expansion of Pi^3/(36sqrt(3)) + 91zeta(3)/216.