A020763 -id:A020763 - cp's OEIS frontend

A179294 Decimal expansion of radius of inscribed sphere about a regular icosahedron with edge = 1.

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Jul 09 2010

Icosahedron: A three-dimensional figure with 20 equilateral triangle faces, 12 vertices, and 30 edges.

			0.75576131407617073048013370202500139263844478889356105922958289203910...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Dr. Math, Icosahedron.
Wikipedia, Icosahedron.
MathWorld, Icosahedron.
Index entries for algebraic numbers, degree 4.

Cf. A010527, A102208, A179290, A179292.

Cf. Platonic solids inradii: A020781 (tetrahedron), A020763 (octahedron), A237603 (dodecahedron).

RealDigits[(Sqrt[42+18Sqrt[5]]/12), 10, 175][[1]]

sqrt((7+3*sqrt(5))/6)/2 \\ Stefano Spezia, Jan 27 2025

Equals sqrt(42 + 18*sqrt(5))/12.

Partially rewritten by Charles R Greathouse IV, Feb 03 2011

A020761 Decimal expansion of 1/2.

Original entry on oeis.org

5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Real part of all nontrivial zeros of the Riemann zeta function (assuming the Riemann hypothesis to be true). - Alonso del Arte, Jul 02 2011
Radius of a sphere with surface area Pi. - Omar E. Pol, Aug 09 2012
Radius of the midsphere (tangent to the edges) in a regular octahedron with unit edges. Also radius of the inscribed sphere (tangent to faces) in a cube with unit edges. - Stanislav Sykora, Mar 27 2014
Construct a rectangle of maximal area inside an arbitrary triangle. The ratio of the rectangle's area to the triangle's area is 1/2. - Rick L. Shepherd, Jul 30 2014

			1/2 = 0.50000000000000...

Michael Penn, A creative approach to a scary looking integral., YouTube video, 2020.
Michael Penn, I really like this sum!, YouTube video, 2021.
Wikipedia, Riemann zeta function
Wikipedia, Platonic solid
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. In platonic solids:
midsphere radii:
A020765 (tetrahedron),
A010503 (cube),
A019863 (icosahedron),
A239798 (dodecahedron);
insphere radii:
A020781 (tetrahedron),
A020763 (octahedron),
A179294 (icosahedron),
A237603 (dodecahedron).

Digits:=100; evalf(1/2); # Wesley Ivan Hurt, Mar 27 2014

RealDigits[1/2, 10, 128][[1]] (* Alonso del Arte, Dec 13 2013 *)
LinearRecurrence[{1},{5,0},99] (* Ray Chandler, Jul 15 2015 *)

{ default(realprecision); x=1/2*10; for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", ")) } \\ Felix Fröhlich, Jul 24 2014

a(n) = 5*(n==0); \\ Michel Marcus, Jul 25 2014

Equals Sum_{k>=1} (1/3^k). Hence 1/2 = 0.1111111111111... in base 3.
Cosine of 60 degrees, i.e., cos(Pi/3).
-zeta(0), zeta being the Riemann function. - Stanislav Sykora, Mar 27 2014
a(0) = 5; a(n) = 0, n > 0. - Wesley Ivan Hurt, Mar 27 2014
a(n) = 5 * floor(1/(n + 1)). - Wesley Ivan Hurt, Mar 27 2014
Equals 2*A019824*A019884. - R. J. Mathar, Jan 17 2021

A157697 Decimal expansion of sqrt(2/3).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Mar 04 2009

Height (from a vertex to the opposite face) of regular tetrahedron with unit edge. - Stanislav Sykora, May 31 2012
The eccentricity of the ellipse of minimum area that is circumscribing two equal and externally tangent circles (Kotani, 1995). - Amiram Eldar, Mar 06 2022
The standard deviation of a roll of a 3-sided die. - Mohammed Yaseen, Feb 23 2023

			0.81649658092772603273242802490196379732198249355222...

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (168) on page 32.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Jisho Kotani, Problem 2053, Crux Mathematicorum, Vol. 5, No. 1 (1995), p. 202; Solution to Problem 2053, by David Hankin, ibid., Vol. 22, No. 4 (1996), pp. 187-188.
D. H. Lehmer, Interesting series involving the Central Binomial Coefficient, Am. Math. Monthly, Vol. 92, No. 7 (1985), pp. 449-457.
Index entries for algebraic numbers, degree 2.

Cf. A002193, A020763, A092259, A115754, A145439.

Sqrt(2/3); // G. C. Greubel, Mar 30 2018

Maple
```
evalf(sqrt(2/3)) ;
```

RealDigits[Sqrt[2/3], 10, 200][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011*)

sqrt(2/3) \\ G. C. Greubel, Mar 30 2018

Equals 1 - (1/2)/2 + (1*3)/(2*4)/2^2 - (1*3*5)/(2*4*6)/2^3 + ... [Jolley]
Equals Sum_{n>=0} (-1)^n*binomial(2n,n)/8^n = 1/A115754. Averaging this constant with sqrt(2) = A002193 = Sum_{n>=0} binomial(2n,n)/8^n yields A145439.
From Michal Paulovic, Dec 08 2022: (Start)
Equals 2 * A020763.
Has periodic continued fraction expansion [0, 1, 4; 2, 4]. (End)
Equals exp(-arctanh(1/5)). - Amiram Eldar, Jul 10 2023
Equals Product_{k>=1} (1 + (-1)^k/A092259(k)). - Amiram Eldar, Nov 24 2024

A237603 Decimal expansion of the inscribed sphere radius in a regular dodecahedron with unit edge.

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Feb 25 2014

Equals phi^2/(2*xi), where phi is the golden ratio (A001622, 2*cos(Pi/5)) and xi is its associate (A182007, 2*sin(Pi/5)).

			1.1135163644116067351943750394869493758831503698864877726012080...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Stanislav Sykora, Table of n, a(n) for n = 1..2000
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A182007, A019863, A019863, A019952, A374771 (sphere volume).

Cf. Platonic solids inradii: A020781 (tetrahedron), A020763 (octahedron), A179294 (icosahedron).

RealDigits[ Cos[Pi/5]^2 / Sin[Pi/5], 10, 111][[1]] (* Or *)
RealDigits[ Sqrt[5/8 + 11/(8 Sqrt[5])], 10, 111][[1]] (* Robert G. Wilson v, Feb 28 2014 *)

PARI
```
sqrt(250+110*sqrt(5))/20
```

Equals A001622^2/A182007 = (cos(Pi/5))^2/sin(Pi/5) = A019863^2/A019845 = cos(Pi/5)*cotan(Pi/5) = A019863*A019952 = 1/sin(Pi/5) - sin(Pi/5) = A019845^(-1) - A019845 = sqrt(250+110*sqrt(5))/20.

A020781 Decimal expansion of 1/sqrt(24).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Radius of the inscribed sphere (tangent to the faces) for a regular tetrahedron with unit edges. - Stanislav Sykora, Nov 20 2013

			1/sqrt(24) = 0.20412414523193150818310700622549094933... . - _Vladimir Joseph Stephan Orlovsky_, May 30 2010

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. Platonic solids inradii: A020763 (octahedron), A179294 (icosahedron), A237603 (dodecahedron). - Stanislav Sykora, Feb 25 2014

Cf. A010464, A010480, A020763, A020853, A021028, A187110, A244980.

RealDigits[N[1/Sqrt[24],200]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

Equals A010464/12. - Stefano Spezia, Jan 26 2025

Equals 1/A010480 = A020763/2 = 2*A020853 = A187110/3 = A244980/Pi. - Hugo Pfoertner, Jan 26 2025

A145439 Decimal expansion of Sum_{k>=0} binomial(4k, 2k)/2^(6*k).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Feb 08 2009

			1.11535507165041054076705837455583093794582718446458...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, 1996, 4.1.49.

Cf. A010503, A020760, A020763, A092259.

Maple
```
1/2*(1+1/3*3^(1/2))*2^(1/2);
```

RealDigits[1/Sqrt[2] + 1/Sqrt[6], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

1/sqrt(6) + 1/sqrt(2) \\ Michel Marcus, Jan 15 2021

Equals (1+A020760)*A010503.
Equals A020763 + A010503. - Artur Jasinski, Dec 20 2020
The minimal polynomial is 9*x^4 - 12*x^2 + 1. - Joerg Arndt, Sep 20 2023
Equals 2F1(1/4,3/4; 1/2; 1/4). - R. J. Mathar, Aug 02 2024
Equals Product_{k>=1} (1 - (-1)^k/A092259(k)). - Amiram Eldar, Nov 24 2024

Typo in definition corrected by R. J. Mathar, Feb 09 2009

A385505 Decimal expansion of the volume of a biaugmented triangular prism with unit edge.

Original entry on oeis.org

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

Offset: 0

Paolo Xausa, Jul 01 2025

The biaugmented triangular prism is Johnson solid J_50.

			0.90441722268325100631575782677970078459225860524491...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Wikipedia, Biaugmented triangular prism.

Cf. A384141 (surface area + 4).

Cf. A020763, A385506.

First[RealDigits[Sqrt[59/144 + 1/Sqrt[6]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J50", "Volume"], 10, 100]]

Equals sqrt(59/144 + 1/sqrt(6)) = sqrt(59/144 + A020763).

Equals the largest root of 20736*x^4 - 16992*x^2 + 25.

A172476 a(n) = floor(n/sqrt(6)).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 17, 17, 17, 18, 18, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26, 27, 27, 28, 28, 28, 29, 29, 30, 30

Offset: 0

Vincenzo Librandi, Feb 04 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Beatty sequences

Cf. A022840: floor(n/(sqrt(6)/6)), Beatty sequence of A020763.

Cf. A194986.

[Floor(n*Sqrt(6)/6): n in [0..80]]; // Vincenzo Librandi, Aug 01 2013

Table[Floor[n Sqrt[6]/6], {n, 0, 80}] (* Vincenzo Librandi, Aug 01 2013 *)

Definition simplified by N. J. A. Sloane, Mar 11 2021

A020799 Decimal expansion of 1/sqrt(42).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

1/sqrt(42) = 0.154303349962091910261094462763999920421552483501389203564556057090705175663... [Vladimir Joseph Stephan Orlovsky, Jun 01 2010]

Ivan Panchenko, Table of n, a(n) for n = 0..1000

RealDigits[N[1/Sqrt[42],200]] (* Vladimir Joseph Stephan Orlovsky, Jun 01 2010 *)

Equals A020763 * A020764 . - R. J. Mathar, Nov 14 2023

cp's OEIS Frontend

A179294 Decimal expansion of radius of inscribed sphere about a regular icosahedron with edge = 1.

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A020761 Decimal expansion of 1/2.

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A157697 Decimal expansion of sqrt(2/3).

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A237603 Decimal expansion of the inscribed sphere radius in a regular dodecahedron with unit edge.

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A020781 Decimal expansion of 1/sqrt(24).

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A145439 Decimal expansion of Sum_{k>=0} binomial(4*k, 2*k)/2^(6*k).

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A145439 Decimal expansion of Sum_{k>=0} binomial(4k, 2k)/2^(6*k).