A010527 -id:A010527 - cp's OEIS frontend

A130590 Decimal expansion of the mean Euclidean distance from a point in the unit 3D cube to a given vertex of the cube.

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

Offset: 0

R. J. Mathar, Aug 10 2007

			0.960591956455052959425107951...

D. H. Bailey, J. M. Borwein and R. E. Crandall, Box Integrals, J. Comp. Appl. Math., Vol. 206, No. 1 (2007), pp. 196-208.
D. H. Bailey, J. M. Borwein, and R. E. Crandall, Advances in the theory of box integrals, Math. Comp. 79 (271) (2010) 1839-1866, Table 2. [From _R. J. Mathar_, Oct 13 2010]
Eric Weisstein's World of Mathematics, Box Integral.

Cf. A010527, A019691, A065914, A135691.

Analogous constants: A244921 (square), A254979 (4-cube).

evalf( sqrt(3)/4+log(2+sqrt(3))/2-Pi/24);

RealDigits[Sqrt[3]/4 + Log[2+Sqrt[3]]/2 - Pi/24, 10, 120][[1]] (* Amiram Eldar, Jun 04 2023 *)

Equals sqrt(3)/4 + log(2+sqrt(3))/2 - Pi/24 = A010527/2 + A065914/2 - A019691.

Equals 2 * A135691. - Amiram Eldar, Jun 04 2023

Name corrected by Amiram Eldar, Jun 04 2023

A179592 Decimal expansion of the circumradius of pentagonal cupola with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 19 2010

Pentagonal cupola: 15 vertices, 25 edges, and 12 faces.

			2.232950509415690049500415383249682772934080730579181647457441260...

Wolfram Alpha, Johnson solid 5
Index entries for algebraic numbers, degree 4.

Cf. A001622, A010527, A102208, A179290, A179292, A179294, A179449, A179450, A179451, A179452, A179552, A179553, A019881, A179587, A179588, A179589, A179590, A179591.

Mathematica
```
RealDigits[N[Sqrt[11+4*Sqrt[5]]/2,200]]
```

Digits of sqrt(11+4*sqrt(5))/2.

A332133 Decimal expansion of (1 + sqrt(3))/2, unique positive root of x^2 - x - 1/2.

Original entry on oeis.org

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

Offset: 1

M. F. Hasler, Oct 29 2020

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

			1.3660254037844386467637231707529361834714026269051903140279...

Kelvin Voskuijl, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2

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

Cf. A001622, A174968.

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

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

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

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

A010153 Continued fraction for sqrt(75) (or 5*sqrt(3)).

Original entry on oeis.org

8, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16, 1, 1, 1, 16

Offset: 0

N. J. A. Sloane

			8.66025403784438646763723170... = 8 + 1/(1 + 1/(1 + 1/(1 + 1/(16 + ...)))). - _Harry J. Smith_, Jun 02 2009

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A010527.

ContinuedFraction[Sqrt[75],300] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2011 *)
PadRight[{8},120,{16,1,1,1}] (* Harvey P. Dale, Oct 05 2024 *)

{ allocatemem(932245000); default(realprecision, 18000); x=contfrac(sqrt(75)); for (n=0, 20000, write("b010153.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 02 2009

From Amiram Eldar, Nov 13 2023: (Start)
Multiplicative with a(2) = 1, a(2^e) = 16 for e >= 2, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 15/4^s). (End)

A092242 Numbers that are congruent to {5, 7} (mod 12).

Original entry on oeis.org

5, 7, 17, 19, 29, 31, 41, 43, 53, 55, 65, 67, 77, 79, 89, 91, 101, 103, 113, 115, 125, 127, 137, 139, 149, 151, 161, 163, 173, 175, 185, 187, 197, 199, 209, 211, 221, 223, 233, 235, 245, 247, 257, 259, 269, 271, 281, 283, 293, 295, 305, 307, 317, 319, 329, 331

Offset: 1

Giovanni Teofilatto, Feb 19 2004

L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 64.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Fifth row of A092260.

Cf. A010527, A120683.

Select[Range[331], MemberQ[{5, 7}, Mod[#, 12]] &] (* Amiram Eldar, Dec 04 2021 *)

1/5^2 + 1/7^2 + 1/17^2 + 1/19^2 + 1/29^2 + 1/31^2 + ... = Pi^2*(2 - sqrt(3))/36 = 0.073459792... [Jolley] - Gary W. Adamson, Dec 20 2006
a(n) = 12*n - a(n-1) - 12 (with a(1)=5). - Vincenzo Librandi, Nov 16 2010
From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 6*n - 3 - 2*(-1)^n.
G.f.: x*(5+2*x+5*x^2) / ( (1+x)*(x-1)^2 ). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (2 - sqrt(3))*Pi/12. - Amiram Eldar, Dec 04 2021
From Amiram Eldar, Nov 24 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sec(Pi/12) (A120683).
Product_{n>=1} (1 + (-1)^n/a(n)) = (sqrt(3)/2)*sec(Pi/12) (= A010527 * A120683). (End)

Edited and extended by Ray Chandler, Feb 21 2004

A093821 Decimal expansion of (2*(3 - sqrt(3)))/3.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Apr 16 2004

Area of lamina found by Pal in the Lebesgue minimal problem.

This appears also as the ratio r_3/r_4 of the outer radii of a regular triangle circumscribed by a regular quadrangle such that they share one vertex and the other two vertices of the triangle touch two sides of the quadrangle. The center of the quadrangle is displaced from the center of the triangle by d/r_4 = 1-2/sqrt(3) = 0.154700.. (see A020832). # R. J. Mathar, Jan 22 2013

			0.845299461...

Eric Weisstein's World of Mathematics, Lebesgue Minimal Problem

Cf. A010527, A093822.

evalf(2*(1-1/sqrt(3))) ; # R. J. Mathar, Jan 22 2013

A165953 Decimal expansion of (5sqrt(3) + sqrt(15))/(6Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Oct 02 2009

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.

			0.6649088942053266431144284467086337161648765805556919381057592605722964718...

Eric Weisstein's World of Mathematics, Dodecahedron.
Index entries for transcendental numbers.

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

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

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

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.

A242703 Decimal expansion of sqrt(7)/2.

Original entry on oeis.org

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

Offset: 1

Alonso del Arte, May 20 2014

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)

			1.32287565553229529525...

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

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

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

{ 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

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

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

A339260 Decimal expansion of the maximum possible volume of a polyhedron with 8 vertices inscribed in the unit sphere.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Nov 29 2020

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.

			1.8157161042244203975084949306331777890131009552754398376663729...

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).
Hugo Pfoertner, Number of edges incident with the 8 vertices, video (2021).

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

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

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

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

A339261 Decimal expansion of the conjecturally maximum possible volume of a polyhedron with 9 vertices inscribed in the unit sphere.

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Dec 05 2020

			2.0437501158996398411663654642269853338632606152947518187182157956871...

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.

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

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

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

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

cp's OEIS Frontend

A130590 Decimal expansion of the mean Euclidean distance from a point in the unit 3D cube to a given vertex of the cube.

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A179592 Decimal expansion of the circumradius of pentagonal cupola with edge length 1.

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A332133 Decimal expansion of (1 + sqrt(3))/2, unique positive root of x^2 - x - 1/2.

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A010153 Continued fraction for sqrt(75) (or 5*sqrt(3)).

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A092242 Numbers that are congruent to {5, 7} (mod 12).

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Mathematica

Formula

Extensions

A093821 Decimal expansion of (2*(3 - sqrt(3)))/3.

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Maple

A165953 Decimal expansion of (5*sqrt(3) + sqrt(15))/(6*Pi).

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PARI

A165953 Decimal expansion of (5sqrt(3) + sqrt(15))/(6Pi).