A104956 -id:A104956 - cp's OEIS frontend

A385578 Decimal expansion of the volume of a parabiaugmented hexagonal prism with unit edge.

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

Offset: 1

Paolo Xausa, Jul 04 2025

The parabiaugmented hexagonal prism is Johnson solid J_55.

Also the volume of a metabiaugmented hexagonal prism (Johnson solid J_56) with unit edge.

			3.0694807321443476232250657536620412432707651725...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Parabiaugmented hexagonal prism.

Cf. A385257 (surface area + 2).

Cf. A002194, A010466, A104956, A131594, A385569.

First[RealDigits[(Sqrt[8] + 9*Sqrt[3])/6, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J55", "Volume"], 10, 100]]

Equals (2*sqrt(2) + 9*sqrt(3))/6 = (A010466 + 9*A002194)/6 = A131594 + A104956.

Equals the largest root of 1296*x^4 - 18072*x^2 + 55225.

A179050 Decimal expansion of 5/(2sqrt(5+2sqrt(5))), area of regular pentagram with base edge length 1.

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jun 26 2010

An algebraic number of degree 4: the smaller positive root of 16x^4 - 200x^2 + 125. - Charles R Greathouse IV, Dec 03 2012

			0.81229924058226581538967853053783616238725867880368775076951179784168...

Thomas M. Green, Area of Pentagram
Index entries for algebraic numbers, degree 4

Cf. A002194, A010502, A102771, A120011, A179047, A179048, A104956.

a=1;area=5/(2*Sqrt[5+2*Sqrt[5]]);RealDigits[N[area,20]]

5/sqrt(20+8*sqrt(5)) \\ Charles R Greathouse IV, Dec 03 2012

Offset corrected, keyword:cons inserted by R. J. Mathar, Jun 28 2010

Name corrected by Charles R Greathouse IV, Dec 03 2012

A212886 Decimal expansion of 2/(3sqrt(3)) = 2sqrt(3)/9.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, May 29 2012

Consider any cubic polynomial f(x) = a(x - r)(x - (r + s))(x -(r + 2s)), where a, r, and s are real numbers with s > 0 and nonzero a; i.e., any cubic polynomial with three distinct real roots, of which the middle root, r + s, is equidistant (with distance s) from the other two. Then the absolute value of f's local extrema is |a|*s^3*(2*sqrt(3)/9). They occur at x = r + s +- s*(sqrt(3)/3), with the local maximum, M, at r + s - s*sqrt(3)/3 when a is positive and at r + s + s*sqrt(3)/3 when a is negative (and the local minimum, m, vice versa). Of course m = -M < 0.
A quadratic number with denominator 9 and minimal polynomial 27x^2 - 4. - Charles R Greathouse IV, Apr 21 2016
This constant is also the maximum curvature of the exponential curve, occurring at the point M of coordinates [x_M = -log(2)/2 = (-1/10)*A016655; y_M = sqrt(2)/2 = A010503]. The corresponding minimum radius of curvature is (3*sqrt(3))/2 = A104956 (see the reference Eric Billault and the link MathStackExchange). - Bernard Schott, Feb 02 2020

			0.384900179459750509672765853667971637098401167513417917345734...

Eric Billault, Walter Damin, Robert Ferréol et al., MPSI - Classes Prépas, Khôlles de Maths, Ellipses, 2012, exercice 17.07 pages 386, 389-390.

MathStackExchange, Apex of an Exponential Function, 2015.
Index entries for algebraic numbers, degree 2

Cf. A002194, A104956, A020760.

Cf. A010503, A016655.

RealDigits[2/(3*Sqrt[3]), 10, 105] (* T. D. Noe, May 31 2012 *)

default(realprecision, 1000); 2*sqrt(3)/9

(2/9)*sqrt(3) = (2/9)*A002194.

A258403 Decimal expansion of the area of the regular 10-gon (decagon) of circumradius = 1.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, May 29 2015

Quartic number of degree 4 and denominator 2; minimal polynomial 16x^4 - 500x^2 + 3125. - Charles R Greathouse IV, Apr 20 2016

			2.9389262614623656458435297731953638429882621882157299553613624...

Chai Wah Wu, Table of n, a(n) for n = 1..10001
Eric Weisstein's MathWorld, Decagon
Wikipedia, Decagon

Cf. A104954 (triangle), A104955 (pentagon), A104956 (hexagon), A104957 (heptagon).

Cf. A178816 (area of decagon with edge length 1). A182007.

RealDigits[(5/2)*Sqrt[(5 - Sqrt[5])/2], 10, 101] // First

(5/2)*sqrt((5 - sqrt(5))/2) \\ Michel Marcus, May 29 2015

Equals (5/2)*sqrt((5-sqrt(5))/2).
Area formulas from triangle to dodecagon, with circumradius 1:
   n-gon    area(n) = (1/2)*n*sin(2*Pi/n)
   3-gon    (3*sqrt(3))/4
   4-gon    2
   5-gon    (5/4)*sqrt((5+sqrt(5))/2)
   6-gon    (3*sqrt(3))/2
   7-gon    (7/2)*cos((3*Pi)/14)
   8-gon    2*sqrt(2)
   9-gon    (9/2)*sin((2*Pi)/9)
  10-gon    (5/2)*sqrt((5-sqrt(5))/2)
  11-gon    (11/2)*sin((2*Pi)/11)
  12-gon    3
This constant is (5/2)*A182007. - Wolfdieter Lang, May 08 2018

A352672 Decimal expansion of r = (3/2)*(1+sqrt(3)).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Mar 26 2022

			4.098076211353315940291169512258808550414...

Cf. A090388, A104956, A352673, A352674, A352676.

Apart from leading digits the same as A176325 and A104956.

r = N[(3/2) (1 + Sqrt[3]), 200]
RealDigits[r][[1]]

Equals A104956 + 3/2. - Michel Marcus, Mar 28 2022

Equals (3/2) * A090388. - Bernard Schott, Mar 28 2022

A208745 Decimal expansion of the gravitoid constant.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Mar 01 2012

Ratio between the width and the depth of the gravitoid curve delimiting any axial section of a gravidome. A gravidome is an axially symmetric homogeneous body shaped in a way to produce, given a constant mass, the maximum possible gravitation field at a point (the barypole) on its surface. It is shaped like a tomato; with respect to a sphere it is somewhat flattened and the gravitoid constant describes the amount of the flattening. The terms "gravidome" for the body and "gravitoid" for its axial perimeter curve were coined in 2006 by S. Sykora.
A quartic number of denominator 3 with minimal polynomial 27x^4 - 64. - Charles R Greathouse IV, Apr 21 2016
Also the diameter from vertex to opposite vertex of the regular hexagon of unit area.  The regular hexagon of unit side has diameter 2 and area (3/2)*sqrt(3) (A104956); scaling that down to unit area yields diameter 2 / sqrt((3/2)*sqrt(3)). - Kevin Ryde, Mar 07 2020

			1.2408064788027994652549583293109787836682723009035365001...

Stanislav Sykora, Ad Astra Ltd and Early History of Gravity Engineering
I. J. Zucker, G. S. Joyce, Special values of the hypergeometric series II, Math. Proc. Cam. Phil. Soc. 131 (2001) 309 (7.6)
Index entries for algebraic numbers, degree 4

RealDigits[2*Sqrt[2/(3*Sqrt[3])],10,120][[1]] (* Harvey P. Dale, Nov 30 2015 *)

2*sqrt(2/3/sqrt(3)) \\ Charles R Greathouse IV, Aug 25 2015

polrootsreal(27*x^4-64)[2] \\ Charles R Greathouse IV, Aug 25 2015

2*sqrt(2/(3*sqrt(3))).
(4/3)^(3/4). - Jon E. Schoenfield, Mar 07 2020
Equals 2F1(1/4,1/2;3/4;3/4) [Zucker] - R. J. Mathar, Jun 24 2024

A333322 Decimal expansion of (3/8) * sqrt(3).

Original entry on oeis.org

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

Offset: 0

Kritsada Moomuang, Mar 15 2020

This is the area of the regular hexagon of diameter 1.
From Bernard Schott, Apr 09 2022 and Oct 01 2022: (Start)
For any triangle ABC, where (A,B,C) are the angles:
  sin(A) * sin(B) * sin(C) <= (3/8) * sqrt(3) [Bottema reference],
  cos(A/2) * cos(B/2) * cos(C/2) <= (3/8) * sqrt(3) [Mitrinovic reference],
and if (ha,hb,hc) are the altitude lengths and (a,b,c) the side lengths of this triangle [Scott Brown link]:
  (ha+hb) * (hb+hc) * (hc+ha) / (a+b) * (b+c) * (c+a) <= (3/8) * sqrt(3).
The equalities are obtained only when triangle ABC is equilateral. (End)

			0.649519052838328985...

O. Bottema et al., Geometric Inequalities, Groningen, 1969, item 2.7, page 19.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.15, p. 526.
D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.2.2, page 111.

Scott Brown, Problem 3453, Crux Mathematicorum, Vol. 36, No. 5 (2010), pp. 342 and 343.

Cf. A002194 (sqrt(3)), A104954.

Cf. A010527, A020821, A104956, A152623 (other geometric inequalities).

RealDigits[(3/8) * Sqrt[3], 10, 120][[1]]

sqrt(27)/8 \\ Charles R Greathouse IV, Apr 09 2022

Equals A104954/2 or A104956/4.

A385694 Decimal expansion of the volume of a triaugmented hexagonal prism with unit edge.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Jul 07 2025

The triaugmented hexagonal prism is Johnson solid J_57.

			3.3051829925398634646920138743636575896990438184040...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Triaugmented hexagonal prism.

Cf. A385259 (surface area + 7).

Cf. A010503, A104956, A385569, A385578.

First[RealDigits[1/Sqrt[2] + 3*Sqrt[3]/2, 10, 100]]
First[RealDigits[PolyhedronData["J57", "Volume"], 10, 100]]

Equals 1/sqrt(2) + 3*sqrt(3)/2 = A010503 + A104956.

Equals the largest root of 16*x^4 - 232*x^2 + 625.

A386435 Decimal expansion of the largest dihedral angle, in radians, in a triangular bipyramid (Johnson solid J_12).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Aug 20 2025

Also the largest dihedral angle in a triangular orthobicupola (Johnson solid J_27) and the second largest dihedral angle in an augmented truncated tetrahedron (Johnson solid J_65).

			2.461918834681549364269858356495974751420680018710...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Wikipedia, Augmented truncated tetrahedron.
Wikipedia, Triangular bipyramid.
Wikipedia, Triangular orthobicupola.

Cf. A137914 (J_12 smallest dihedral angle).

Cf. A020775 (J_12 volume), A104956 (J_12 surface area).

First[RealDigits[ArcCos[-7/9], 10, 100]] (* or *)
First[RealDigits[Max[PolyhedronData["J12", "DihedralAngles"]], 10, 100]]

Equals arccos(-7/9).

A370562 Decimal expansion of (2Pi - 3sqrt(3))/2.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Mar 15 2024

This constant is the difference of the area of a disk with radius 1 (length unit) and the inscribed regular hexagon.

			0.5435164422364772981714738710206943337829615186595348788912...

Cf. A000796, A019692, A010482, A104956, A248897.

RealDigits[Pi - 3*Sqrt[3]/2, 10, 120][[1]] (* Amiram Eldar, Mar 15 2024 *)

Equals (A019692 - A010482)/2.

Equals Pi - 3*sqrt(3)/2 = A000796 - A104956.

cp's OEIS Frontend

A385578 Decimal expansion of the volume of a parabiaugmented hexagonal prism with unit edge.

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A179050 Decimal expansion of 5/(2*sqrt(5+2*sqrt(5))), area of regular pentagram with base edge length 1.

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A212886 Decimal expansion of 2/(3*sqrt(3)) = 2*sqrt(3)/9.

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A258403 Decimal expansion of the area of the regular 10-gon (decagon) of circumradius = 1.

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A352672 Decimal expansion of r = (3/2)*(1+sqrt(3)).

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A208745 Decimal expansion of the gravitoid constant.

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A333322 Decimal expansion of (3/8) * sqrt(3).

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A179050 Decimal expansion of 5/(2sqrt(5+2sqrt(5))), area of regular pentagram with base edge length 1.

A212886 Decimal expansion of 2/(3sqrt(3)) = 2sqrt(3)/9.

A370562 Decimal expansion of (2Pi - 3sqrt(3))/2.