A020761 -id:A020761 - cp's OEIS frontend

A235362 Decimal expansion of the cube root of 2 divided by 2.

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

Offset: 0

Alonso del Arte, Jan 07 2014

Also reciprocal of the real cubic root of 4 and negated real part of either complex cubic root of 2.

			0.6299605249474365823836053...

Richard Zippel, The Weyl Computer Algebra Substrate in Design and Implementation of Symbolic Computation Systems: International Symposium, DISCO '93 Gmunden, Austria, September 15-17, 1993. Proceedings, p. 306.

Cf. A002580, A010503, A020761, A239797.

Digits := 100 ; evalf(1/2^(2/3)) ; # R. J. Mathar, Jan 16 2023

Mathematica
```
RealDigits[1/2^(2/3), 10, 128][[1]]
```

sqrtn(1/4,3) \\ Charles R Greathouse IV, Apr 14 2014

2^(1/3)/2 = 1/2^(2/3) = 1/4^(1/3).
(-2^(1/3)/2 + sqrt(-3)/4^(1/3))^3 = 2.
Equals 1/A005480 = A002580 /2 . - Wolfdieter Lang, Jan 02 2023

A270714 Decimal expansion of (1/2)^(1/3).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling, Mar 22 2016

Let c = (1/2)^(1/3). A sphere of radius c*r has half the volume of a sphere of radius r. - Rick L. Shepherd, Aug 12 2016

Let c = (1/2)^(1/3). The relative maximum of xy(x+y)=1 is (c,-1/c^2). - Clark Kimberling, Oct 05 2020

			0.79370052598409973737585281963615413019574666394992650490414288091260825...

Clark Kimberling, Table of n, a(n) for n = 0..10000

Cf. A002580, A005480, A235362, A269993.

Mathematica
```
RealDigits[(1/2)^(1/3), 10, 200][[1]]
```
PARI
```
(1/2)^(1/3) \\ Altug Alkan, Mar 22 2016
```

Equals 1/A002580 = A002580*A235362 = A005480*A020761. [corrected and expanded by Rick L. Shepherd, Aug 12 2016]

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

A281065 Decimal expansion of the greatest minimum separation between ten points in a unit square.

Original entry on oeis.org

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

Offset: 0

Jeremy Tan, Jan 14 2017

The corresponding values for two to nine points have simple expressions:
  N ... d_min
  2 ... sqrt(2)            (A002193)
  3 ... sqrt(6) - sqrt(2)  (A120683)
  4 ... 1                  (A000007)
  5 ... sqrt(2) / 2        (A010503)
  6 ... sqrt(13) / 6       (A381485)
  7 ... 4 - 2*sqrt(3)      (A379338)
  8 ... sqrt(2 - sqrt(3))  (A101263)
  9 ... 1 / 2              (A020761)
In contrast, the value for ten points has a minimal polynomial of degree 18.
The smallest square ten unit circles will fit into has side length s = 2 + 2/d = 6.74744152... and the maximum radius of ten non-overlapping circles in the unit square is 1 / s = 0.14820432...

			0.421279543983903432768821760650298...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

C. de Groot, R. Peikert, and D. Würtz, The Optimal Packing of Ten Equal Circles in a Square, IPS Research Report, ETH Zürich, No. 90-12, August 1990.
Eckard Specht, The best known packings of equal circles in a square.
Jeremy Tan, Sympy (Python) program.
Index entries for algebraic numbers, degree 18.

Cf. A281115 (10 points in unit circle), A000007, A002193, A010503, A020761, A101263, A120683, A379338, A381485.

RealDigits[x /. FindRoot[x^Range[18, 0, -1].{1180129, -11436428, 98015844, -462103584, 1145811528, -1398966480, 227573920, 1526909568, -1038261808, -2960321792, 7803109440, -9722063488, 7918461504, -4564076288, 1899131648, -563649536, 114038784, -14172160, 819200}, {x, 2/5}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Feb 24 2025 *)

my(p = Pol([1180129, -11436428, 98015844, -462103584, 1145811528, -1398966480, 227573920, 1526909568, -1038261808, -2960321792, 7803109440, -9722063488, 7918461504, -4564076288, 1899131648, -563649536, 114038784, -14172160, 819200])); polrootsreal(p)[1]

d is the smallest real root of 1180129*d^18 - 11436428*d^17 + 98015844*d^16 - 462103584*d^15 + 1145811528*d^14 - 1398966480*d^13 + 227573920*d^12 + 1526909568*d^11 - 1038261808*d^10 - 2960321792*d^9 + 7803109440*d^8 - 9722063488*d^7 + 7918461504*d^6 - 4564076288*d^5 + 1899131648*d^4 - 563649536*d^3 + 114038784*d^2 - 14172160*d + 819200.

A341906 Decimal expansion of the moment of inertia of a solid regular dodecahedron with a unit mass and a unit edge length.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 04 2021

The moments of inertia of the five Platonic solids were apparently first calculated by the Canadian physicist John Satterly (1879-1963) in 1957.
The moment of inertia of a solid regular dodecahedron with a uniform mass density distribution, mass M, and edge length L is I = c*M*L^2, where c is this constant.
The corresponding values of c for the other Platonic solids are:
Tetrahedron: 1/20 (= A020761/10).
Octahedron: 1/10 (= A000007).
Cube: 1/6 (= A020793).
Icosahedron: (3 + sqrt(5))/20 (= A104457/10).

			0.60735550374163932719985924360173257727394705341616...

P. K. Aravind, Gravitational collapse and moment of inertia of regular polyhedral configurations, American Journal of Physics, Vol. 59, No. 7 (1991), pp. 647-652.
Frédéric Perrier, Moments of inertia of Archimedean solids, 2015.
John Satterly, Moments of Inertia about Selected Axes of Regular Polygons, Right Pyramids on Regular Polygonal Bases, and of the Platonic and Some Archimedian Polyhedra, American Journal of Physics, Vol. 25, No. 7 (1957), pp. 489-490.
John Satterly, The Moments of Inertia of Some Polyhedra, The Mathematical Gazette, Vol. 42, No. 339 (1958), pp. 11-13.
Wikipedia, List of moments of inertia.
Wikipedia, Moment of inertia.
Wikipedia, Regular dodecahedron.
Index entries for sequences related to moment of inertia.

Cf. A000007, A001622, A020761, A020793, A104457.

Other constants related to the regular dodecahedron: A102769, A131595, A179296, A232810, A237603, A239798.

RealDigits[(95 + 39*Sqrt[5])/300, 10, 100][[1]]

Equals (95 + 39*sqrt(5))/300.

Equals (28 + 39*phi)/150, where phi is the golden ratio (A001622).

A069181 Decimal expansion of 1/1024.

Original entry on oeis.org

0, 0, 0, 9, 7, 6, 5, 6, 2, 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

Offset: 0

Rick L. Shepherd, Apr 10 2002

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

Cf. A021516 (1/512), A021260 (1/256), A021132 (1/128), A021068 (1/64), A021036 (1/32), A021020 (1/16), A020821 (1/8), A020773 (1/4), A020761 (1/2).

Join[{0,0,0},RealDigits[1/1024,10,120][[1]]] (* or *) PadRight[ {0,0,0,9,7,6,5,6,2,5},120,{0}] (* Harvey P. Dale, Jan 26 2019 *)

A214174 Decimal expansion of Integral_{x=0..oo} x/cosh(Pi*x) dx.

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Mar 20 2013

This sequence is between A020761 [Integral_{x=0..oo} x^0/cosh(Pi*x) dx] and A020821 [Integral_{x=0..oo} x^2/cosh(Pi*x) dx].

			0.1856134363553860814769819901997199856094304591312932054076019...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A006752, A020761, A020821, A060294, A143233.

SetDefaultRealField(RealField(100)); R:=RealField(); 2*Catalan(R)/Pi(R)^2; // G. C. Greubel, Aug 25 2018

Mathematica
```
RealDigits[2 Catalan/Pi^2, 10, 90][[1]]
```

default(realprecision, 100); 2*Catalan/Pi^2 \\ G. C. Greubel, Aug 25 2018

Equals A060294*A143233 = 2*C/Pi^2, where C is Catalan's constant (A006752).

A358981 Decimal expansion of Pi/3 - sqrt(3)/4.

Original entry on oeis.org

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

Offset: 0

Michal Paulovic, Dec 08 2022

The constant is the area of a circular segment bounded by an arc of 2*Pi/3 radians (120 degrees) of a unit circle and by a chord of length sqrt(3). Three such segments result when an equilateral triangle with side length sqrt(3) is circumscribed by a unit circle. The area of each segment is:
  A = (R^2 / 2) * (theta - sin(theta))
  A = (1^2 / 2) * (2*Pi/3 - sin(2*Pi/3))
  A = (1 / 2) * (2*Pi/3 - sqrt(3)/2)
  A = Pi/3 - sqrt(3)/4 = (Pi - 3*sqrt(3)/4) / 3 = 0.61418484...
where Pi (A000796) is the area of the circle, and 3*sqrt(3)/4 (A104954) is the area of the inscribed equilateral triangle.
The sagitta (height) of the circular segment is:
  h = R * (1 - cos(theta/2))
  h = 1 * (1 - cos(Pi/3))
  h = 1 - 1/2 = 0.5 (A020761)

			0.6141848493043784...

Index entries for transcendental numbers

Cf. A000796, A019670, A020761, A093731, A104954, A120011.

Maple
```
evalf(Pi/3-sqrt(3)/4);
```

RealDigits[Pi/3 - Sqrt[3]/4, 10, 100][[1]]

PARI
```
Pi/3 - sqrt(3)/4
```

Equals A019670 - A120011. - Omar E. Pol, Dec 08 2022

Equals A093731 / 2. - Michal Paulovic, Mar 08 2024

A381485 Decimal expansion of sqrt(13)/6.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Feb 24 2025

The greatest possible minimum distance between 6 points in a unit square.

The solution was found by Ronald L. Graham and reported by Schaer (1965).

			0.60092521257733154885320354457841599104188276230754...

Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved Problems in Geometry, Springer, 1991, Section D1, p. 108.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, Vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

J. Schaer, The densest packing of 9 circles in a square, Canadian Mathematical Bulletin, Vol. 8, No. 3 (1965), pp. 273-277.
D. Würtz, M. Monagan, and R. Peikert, The history of packing circles in a square, Maple Technical Newsletter, Vol. 1 (1994), pp. 35-42; ResearchGate link.
Index entries for algebraic numbers, degree 2.

Solutions for k points: A002193 (k = 2), A120683 (k = 3), 1 (k = 4), A010503 (k = 5), this constant (k = 6), A379338 (k = 7), A101263 (k = 8), A020761 (k = 9), A281065 (k = 10).

Cf. A010470, A142464, A176019, A295330, A344069.

Mathematica
```
RealDigits[Sqrt[13] / 6, 10, 120][[1]]
```

list(len) = digits(floor(10^len*quadgen(52)/6));

Equals A010470 / 6 = A295330 / 3 = A344069 / 2 = A176019 - 1/2 = sqrt(A142464).

Minimal polynomial: 36*x^2 - 13.

A021108 Decimal expansion of 1/104.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

After 9, periodic with period 6: [6, 1, 5, 3, 8, 4]. See also A021030 (1/26), A021069 (1/65), A021420 (1/416), A021654 (1/650). - Bruno Berselli, Apr 13 2018

			0.009615384615384615384615384615384615384615384615384615384615384...

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

Join[{0, 0}, RealDigits[1/104, 10, 120][[1]]] (* or *) PadRight[{0, 0, 9, 6}, 120,{3, 8, 4, 6, 1, 5}] (* Harvey P. Dale, Aug 18 2012 *)

Equals A020821 * A021017 = A020773 * A021030 = A020761 * A021056. - Bruno Berselli, Apr 13 2018

A335094 Decimal expansion of (15 - 4*sqrt(2))/8.

Original entry on oeis.org

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

Offset: 1

Jeremy Tan, Sep 12 2020

Largest overhang off an edge achievable with four unit-length bricks.

Corresponding values for one, two and three bricks are 1/2, 3/4 and 1 respectively.

			1.1678932188134524755991556378951...

S. Ainley, Finely Balanced, The Mathematical Gazette, 63 (1979), p. 272.
M. Paterson and U. Zwick, Overhang, arXiv:0710.2357 [math.HO], 2007.
Eric Weisstein's World of Mathematics, Book Stacking Problem

Cf. A020761 (1/2), A152627 (3/4).

Cf. A014537, A157215, A268682.

First@ RealDigits@ N[(15 - 4 Sqrt[2])/8, 102]

PARI
```
(15-4*sqrt(2))/8
```

cp's OEIS Frontend

A235362 Decimal expansion of the cube root of 2 divided by 2.

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A270714 Decimal expansion of (1/2)^(1/3).

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A281065 Decimal expansion of the greatest minimum separation between ten points in a unit square.

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A341906 Decimal expansion of the moment of inertia of a solid regular dodecahedron with a unit mass and a unit edge length.

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A069181 Decimal expansion of 1/1024.

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A214174 Decimal expansion of Integral_{x=0..oo} x/cosh(Pi*x) dx.

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Magma

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A358981 Decimal expansion of Pi/3 - sqrt(3)/4.

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