A010527 -id:A010527

A165953

Decimal expansion of (5*sqrt(3) + sqrt(15))/(6*Pi).

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

The ratio of the volume of a regular dodecahedron to the volume of the circumscribed sphere (which has circumradius a*(sqrt(3) + sqrt(15))/4 = a*(A002194 + A010472)/4, where a is the dodecahedron's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165952, and A165954. A063723 shows the order of these by size.

LINKS

Table of n, a(n) for n=0..104.

Eric Weisstein's World of Mathematics, Dodecahedron.

Index entries for transcendental numbers.

FORMULA

Equals (5*A002194 + A010472)/(6*A000796).

Equals (5*A002194 + A010472)*A049541/6.

Equals (10*A010527 + A010472)*A049541/6.

Equals (5 + sqrt(5))/(2*Pi*sqrt(3)).

Equals (5 + A002163)*A049541*A020760/2.

EXAMPLE

0.6649088942053266431144284467086337161648765805556919381057592605722964718...

MATHEMATICA

RealDigits[(5*Sqrt[3]+Sqrt[15])/(6*Pi), 10, 120][[1]] (* Harvey P. Dale, Feb 16 2018 *)

PROG

(PARI) (5*sqrt(3)+sqrt(15))/(6*Pi)

CROSSREFS

Cf. A000796, A002194, A010472, A165922, A049541, A165952, A165954, A063723, A002163, A020760, A010527.

KEYWORD

cons,nonn

AUTHOR

Rick L. Shepherd, Oct 02 2009

STATUS

approved

A179637

Decimal expansion of the surface area of pentagonal rotunda with edge length 1.

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

2,1

COMMENTS

Pentagonal rotunda: 20 vertices, 35 edges, and 17 faces.

LINKS

Table of n, a(n) for n=2..106.

Wolfram Alpha, Johnson solid 6

FORMULA

Digits of sqrt(5*(145+58*sqrt(5)+2*sqrt(30*(65+29*sqrt(5)))))/2.

EXAMPLE

22.3472002653941282767984141581886130738180135134316226129799763167102...

MATHEMATICA

RealDigits[N[Sqrt[5*(145+58*Sqrt[5]+2*Sqrt[30*(65+29*Sqrt[5])])]/2, 200]]

CROSSREFS

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449-A179452, A179552-A179554, A179587-A179593.

KEYWORD

cons,nonn,easy

AUTHOR

Vladimir Joseph Stephan Orlovsky, Jul 21 2010

EXTENSIONS

Offset corrected by R. J. Mathar, Aug 15 2010

STATUS

approved

A242703

Decimal expansion of sqrt(7)/2.

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Absolute value of the imaginary part of any of the nontrivial divisors of 2 in O_Q(sqrt(-7)).

The incircle of a triangle with sides of lengths 4, 5, 6 units respectively has a radius of sqrt(7)/2.

With a different offset, decimal expansion of 5 * sqrt(7).

From Wolfdieter Lang, Nov 18 2017: (Start)

In a regular hexagon inscribed in a circle with a radius of 1 unit the three distinct distances between any vertex and the middle of the sides are 1/2, sqrt(7)/2 and sqrt(13)/2.

The continued fraction expansion of sqrt(7)/2 is 1, repeat(3, 10, 3, 2). The convergents are given in A294972/A294973. (End)

LINKS

Iain Fox, Table of n, a(n) for n = 1..20000 (first 1000 terms from Ivan Panchenko)

FORMULA

(1/2 - sqrt(-7)/2)(1/2 + sqrt(-7)/2) = 2.

Equals A010465/2. - R. J. Mathar, Feb 23 2021

EXAMPLE

1.32287565553229529525...

MATHEMATICA

RealDigits[Sqrt[7]/2, 10, 100][[1]]

PROG

(PARI) { default(realprecision, 20080); x=sqrt(7)/2; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b242703.txt", n, " ", d)); } \\ Iain Fox, Nov 18 2017

CROSSREFS

Cf. A010527, A020837, A040022, A294972/A294973.

KEYWORD

easy,cons,nonn

AUTHOR

Alonso del Arte, May 20 2014

STATUS

approved

A332133

Decimal expansion of (1 + sqrt(3))/2, unique positive root of x^2 - x - 1/2.

+10
4

1, 3, 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, 3, 3, 7, 8, 6, 2, 4, 2, 8, 7, 8, 3, 7, 8

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Also, max {a, b} where {a,b} is the unique solution of a + b = 1 and a^2 + b^2 = 2 (implying also ab = -1/2 and a^3 + b^3 = 5/2 without solving for a, b). See A332122 for a generalization to 3 variables {a, b, c}.

This is a non-integer element of the quadratic number field Q(sqrt(3)) with the given monic minimal polynomial. The other negative root is -(-1 + sqrt(3))/2 = - A152422. - Wolfdieter Lang, Aug 30 2022

LINKS

Table of n, a(n) for n=1..104.

Index entries for algebraic numbers, degree 2

FORMULA

Equals 1/2 + Sum_{n>=0} ((-1)^(n + 1)*binomial(2*n, n))/(2^(3*n + 1/2)*(2*n - 1)). - Antonio Graciá Llorente, Nov 11 2024

EXAMPLE

1.3660254037844386467637231707529361834714026269051903140279...

MATHEMATICA

RealDigits[(1 + Sqrt[3])/2, 10, 120][[1]] (* Amiram Eldar, Jun 21 2023 *)

PROG

(PARI) localprec(111); digits(solve(a=0, 2, a^2-a-1/2)\.1^99)

(PARI) polrootsreal(2*x^2-2*x-1)[2] \\ Charles R Greathouse IV, Jan 26 2023

CROSSREFS

Cf. A152422 (this - 1 = (sqrt(3)-1)/2), A010527, A332122 (analog for 3rd degree).

Cf. A001622, A174968.

KEYWORD

nonn,cons

AUTHOR

M. F. Hasler, Oct 29 2020

STATUS

approved

A339260

Decimal expansion of the maximum possible volume of a polyhedron with 8 vertices inscribed in the unit sphere.

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Berman and Hanes (see link, page 81) proved in 1970 that an arrangement of 8 points on the surface of a sphere with 4 points with node degree 4 and 4 points with node degree 5 is the one with a maximum volume of their convex hull.

LINKS

Table of n, a(n) for n=1..87.

Joel D. Berman and Kitt Hanes, Volumes of Polyhedra Inscribed in the Unit Sphere in E3. Mathematische Annalen 188, 78-84 (1970).

Donald W. Grace, Search For Largest Polyhedra, Mathematics of Computation 17 (1963), pp. 197-199.

Matt Parker, The search for the biggest shape in the universe, YouTube video, 2024.

Hugo Pfoertner, Visualization of Polyhedron, (1999).

FORMULA

Equals sqrt((475 + 29*sqrt(145))/250).

EXAMPLE

1.8157161042244203975084949306331777890131009552754398376663729...

MATHEMATICA

RealDigits[Sqrt[(475 + 29*Sqrt[145])/250], 10, 120][[1]] (* Amiram Eldar, Jun 01 2023 *)

PROG

(PARI) sqrt((475+29*sqrt(145))/250)

CROSSREFS

Cf. A010527 (volume of double 5-pyramid), A081314, A081366, A122553 (volume of octahedron), A339259.

KEYWORD

nonn,cons

AUTHOR

Hugo Pfoertner, Nov 29 2020

STATUS

approved

A339261

Decimal expansion of the conjecturally maximum possible volume of a polyhedron with 9 vertices inscribed in the unit sphere.

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Table of n, a(n) for n=1..87.

R. H. Hardin, N. J. A. Sloane and W. D. Smith, Maximal Volume Spherical Codes.

Hugo Pfoertner, Visualization of Polyhedron, (1999).

Hugo Pfoertner, 9-Vertex-Polyhedron with maximum volume inscribed in a sphere, YouTube video, Feb 10 2021.

FORMULA

Equals 3*sqrt(2*sqrt(3) - 3).

EXAMPLE

2.0437501158996398411663654642269853338632606152947518187182157956871...

MATHEMATICA

RealDigits[3*Sqrt[2*Sqrt[3] - 3], 10, 120][[1]] (* Amiram Eldar, Jun 28 2023 *)

PROG

(PARI) 3*sqrt(2*sqrt(3) - 3)

CROSSREFS

Cf. A010527 (volume of double 5-pyramid), A081314, A081366, A122553 (volume of octahedron), A339259, A339260, A339261, A339262, A339263.

KEYWORD

nonn,cons

AUTHOR

Hugo Pfoertner, Dec 05 2020

STATUS

approved

A339262

Decimal expansion of the conjecturally maximum possible volume of a polyhedron with 10 vertices inscribed in the unit sphere.

+10
4

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The polyhedron (see linked illustration) has vertices at the poles and two square rings of vertices rotated by Pi/4 against each other, with a polar angle of approx. +-62.89908285 degrees against the poles. The polyhedron is completely described by this angle and its order 16 symmetry. It would be desirable to know a closed formula representation of this angle and the volume.

LINKS

Table of n, a(n) for n=1..87.

R. H. Hardin, N. J. A. Sloane and W. D. Smith, Maximal Volume Spherical Codes.

Hugo Pfoertner, Visualization of Polyhedron, (1999).

EXAMPLE

2.218711131545399403247282751128417013810725374663344381750049084201...

CROSSREFS

Cf. A010527 (volume of double 5-pyramid), A081314, A081366, A122553 (volume of octahedron), A339259, A339260, A339261, A339263.

KEYWORD

nonn,cons

AUTHOR

Hugo Pfoertner, Dec 07 2020

STATUS

approved

A126664

Continued fraction expansion of sqrt(3)/2.

+10
3

0, 1, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6, 2, 6

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Also continued fraction expansion of sin(Pi/3).

LINKS

Table of n, a(n) for n=0..110.

Index entries for linear recurrences with constant coefficients, signature (0,1).

FORMULA

a(n) = 2*(2+(-1)^n) for n>1. a(n) = a(n-2) for n>3. G.f.: -x*(x^2+6*x+1) / ((x-1)*(x+1)). - Colin Barker, Jan 10 2014

MATHEMATICA

ContinuedFraction[Sqrt[3]/2, 120] (* or *) PadRight[{0, 1}, 120, {6, 2}] (* Harvey P. Dale, May 14 2016 *)

CROSSREFS

Cf. A010527.

KEYWORD

cofr,nonn,easy

AUTHOR

Zerinvary Lajos, Feb 10 2007

EXTENSIONS

More terms from N. J. A. Sloane, Dec 29 2008

STATUS

approved

A152422

Decimal expansion of (sqrt(3)-1)/2.

+10
3

3, 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, 3, 3, 7, 8, 6, 2, 4, 2, 8, 7

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

The number has continued fraction [0, 2, 1, 2, 1, 2, 1, ...].

The iterated function z^2 - 1/2 gives a good rational approximation of this number times -1 after sixty steps. - Alonso del Arte, Apr 10 2016

LINKS

Table of n, a(n) for n=0..98.

FORMULA

Equals Product_{k>=1} (1 + (-1)^k * 2/(6*k-3)). - Amiram Eldar, Aug 10 2020

EXAMPLE

0.36602540378443864676372317...

MAPLE

Digits:=100; evalf((sqrt(3)-1)/2); # Wesley Ivan Hurt, Apr 18 2014

MATHEMATICA

RealDigits[(Sqrt[3] - 1)/2, 10, 100] [[1]] (* Wesley Ivan Hurt, Apr 18 2014 *)

PROG

(PARI) (sqrt(3)-1)/2 \\ Altug Alkan, Apr 11 2016

CROSSREFS

A variant of A010527, which is the main entry. - N. J. A. Sloane, Dec 04 2008

KEYWORD

nonn,cons,easy

AUTHOR

Geoffrey Caveney, Dec 03 2008

EXTENSIONS

a(98) corrected by Georg Fischer, Apr 03 2020

STATUS

approved

A339263

Decimal expansion of the conjecturally maximum possible volume of a polyhedron with 11 vertices inscribed in the unit sphere.

+10
3

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The polyhedron (see linked illustration) with a symmetry group of order 4 has a vertex in the north pole on its axis of symmetry. The remaining 10 vertices are diametrically opposite in pairs relative to this axis of symmetry. The polar vertex has vertex degree 6. 8 vertices have vertex degree 5. 2 vertices have vertex degree 4.

This allocation seems to be the best possible approximation of a medial distribution of the vertex degrees, which is a known necessary condition for maximum volume. Of the 25 possible triangulations with vertex degree >= 4, all the others have more than 2 vertices with vertex degree 4, which leads to more pointed corners and therefore smaller volumes.

LINKS

Table of n, a(n) for n=1..87.

R. H. Hardin, N. J. A. Sloane and W. D. Smith, Maximal Volume Spherical Codes.

Hugo Pfoertner, Visualization of Polyhedron, (1999).

EXAMPLE

2.35463449506861520323688059263892654160344864269342168599607566...

CROSSREFS

Cf. A010527 (volume of double 5-pyramid), A081314, A081366, A122553 (volume of octahedron), A339259, A339260, A339261, A339262.

KEYWORD

nonn,cons

AUTHOR

Hugo Pfoertner, Dec 07 2020

STATUS

approved