A010474 -id:A010474 - cp's OEIS frontend

A384142 Decimal expansion of the volume of a gyroelongated square bipyramid with unit edge.

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

Offset: 1

Paolo Xausa, May 22 2025

The gyroelongated square bipyramid is Johnson solid J_17.

			1.428404502627748400527146549078867927980904164...

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

Cf. A010502 (surface area).

Cf. A002193, A010474, A179638.

First[RealDigits[(Sqrt[2] + Sqrt[4 + Sqrt[18]])/3, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J17", "Volume"], 10, 100]]

Equals (sqrt(2) + sqrt(4 + 3*sqrt(2)))/3 = (A002193 + sqrt(4 + A010474))/3.

Equals the largest real root of 81*x^4 - 108*x^2 - 72*x-14.

A011014 Decimal expansion of 4th root of 18.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			2.0597671439071177558302772552010107801...

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A010474, A010590.

RealDigits[18^(1/4), 10, 100][[1]] (* Alonso del Arte, Nov 18 2016 *)

sqrtn(18,4) \\ Charles R Greathouse IV, Apr 15 2014

A011395 Decimal expansion of 6th root of 18.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			1.61887040686056665193034800527...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Index entries for algebraic numbers, degree 6

Cf. A010474, A010590.

RealDigits[18^(1/6), 10, 100][[1]] (* Alonso del Arte, Nov 18 2016 *)

sqrtn(18,6) \\ Charles R Greathouse IV, Nov 26 2024

Last two digits corrected by Ivan Panchenko, Sep 06 2014

A017959 Powers of sqrt(18) rounded to nearest integer.

Original entry on oeis.org

1, 4, 18, 76, 324, 1375, 5832, 24743, 104976, 445375, 1889568, 8016758, 34012224, 144301645, 612220032, 2597429617, 11019960576, 46753733110, 198359290368, 841567195983, 3570467226624, 15148209527701

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A010474, A001027 (bisection).

[Round(Sqrt(18)^n): n in [0..30]]; // Vincenzo Librandi, Nov 20 2011

Digits:=1500: A017959:=n->round(sqrt(18)^n): seq(A017959(n), n=0..30); # Wesley Ivan Hurt, Apr 18 2017

Floor[(Sqrt[18])^Range[0,25]+1/2] (* Vincenzo Librandi, Nov 20 2011 *)

a(n)=round(sqrt(18)^n) \\ Charles R Greathouse IV, Nov 18 2011

A358614 Decimal expansion of 9*sqrt(2)/32.

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Dec 05 2022

Smallest constant M such that the inequality
|a*b*(a^2 - b^2) + b*c*(b^2 - c^2) + c*a*(c^2 - a^2)| <= M * (a^2 + b^2 + c^2)^2
holds for all real numbers a, b, c.
Equality stands for any triple (a, b, c) proportional to (1 - 3*sqrt(2)/2, 1, 1 + 3*sqrt(2)/2), up to permutation.
This constant is the answer to the 3rd problem, proposed by Ireland during the 47th International Mathematical Olympiad in 2006 at Ljubljana, Slovenia (see links).
Equivalently |(a - b)(b - c)(c - a)(a + b + c)| / (a^2 + b^2 + c^2)^2 <= M  with (a,b,c) != (0,0,0).

			0.3977475644174329824...

Evan Chen, IMO 2006/3, IMO 2006 Solution Notes.
The IMO compendium, Problem 3, 47th IMO 2006.
Index to sequences related to Olympiads.

Cf. A002193, A010474, A010503, A230981.

Maple
```
evalf(9*sqrt(2)/32), 100);
```

RealDigits[9*Sqrt[2]/32, 10, 120][[1]] (* Amiram Eldar, Dec 05 2022 *)

Equals (3/16) * A230981 = (3/32) * A010474 = (9/32) * A002193 = (9/16) * A010503.

A379469 Decimal expansion of (1 + sqrt(6))/(3*sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 23 2024

This constant gives an upper bound to the Steiner ratio of a regular tetrahedron.

			0.81305252958514160597609690120357380207588039716628...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.6, p. 505.

Ding-Zhu Du and Warren D. Smith, Disproofs of Generalized Gilbert-Pollak Conjecture on the Steiner Ratio in Three or More Dimensions, Journal of Combinatorial Theory, Series A, Volume 74, Issue 1, 1996, Pages 115-130. See p. 128.
Warren D. Smith, How to find Steiner minimal trees in euclidean d-space, Algorithmica 7, 137-177 (1992).

Cf. A002193, A010464, A010474, A086180.

RealDigits[(1+Sqrt[6])/(3Sqrt[2]),10,100][[1]]

Equals A086180/A010474.

Minimal polynomial: 324*x^4 - 252*x^2 + 25. - Stefano Spezia, Aug 03 2025

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A384142 Decimal expansion of the volume of a gyroelongated square bipyramid with unit edge.

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A011014 Decimal expansion of 4th root of 18.

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A011395 Decimal expansion of 6th root of 18.

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A017959 Powers of sqrt(18) rounded to nearest integer.

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A358614 Decimal expansion of 9*sqrt(2)/32.

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A379469 Decimal expansion of (1 + sqrt(6))/(3*sqrt(2)).

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