cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A010527 Decimal expansion of sqrt(3)/2.

Original entry on oeis.org

8, 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, 1, 3
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

This is the ratio of the height of an equilateral triangle to its base.
Essentially the same sequence arises from decimal expansion of square root of 75, which is 8.6602540378443864676372317...
Also the real part of i^(1/3), the cubic root of i. - Stanislav Sykora, Apr 25 2012
Gilbert & Pollak conjectured that this is the Steiner ratio rho_2, the least upper bound of the ratio of the length of the Steiner minimal tree to the length of the minimal tree in dimension 2. (See Ivanov & Tuzhilin for the status of this conjecture as of 2012.) - Charles R Greathouse IV, Dec 11 2012
Surface area of a regular icosahedron with unit edge is 5*sqrt(3), i.e., 10 times this constant. - Stanislav Sykora, Nov 29 2013
Circumscribed sphere radius for a cube with unit edges. - Stanislav Sykora, Feb 10 2014
Also the ratio between the height and the pitch, used in the Unified Thread Standard (UTS). - Enrique Pérez Herrero, Nov 13 2014
Area of a 30-60-90 triangle with shortest side equal to 1. - Wesley Ivan Hurt, Apr 09 2016
If a, b, c are the sides of a triangle ABC and h_a, h_b, h_c the corresponding altitudes, then (h_a+h_b+h_c) / (a+b+c) <= sqrt(3)/2; equality is obtained only when the triangle is equilateral (see Mitrinovic reference). - Bernard Schott, Sep 26 2022

Examples

			0.86602540378443864676372317...

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 8.2, 8.3 and 8.6, pp. 484, 489, and 504.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), pp. 450-451.
  • D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, and J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.8, page 114.

Links

Crossrefs

Cf. A010153.
Cf. Platonic solids surfaces: A002194 (tetrahedron), A010469 (octahedron), A131595 (dodecahedron).
Cf. Platonic solids circumradii: A010503 (octahedron), A019881 (icosahedron), A179296 (dodecahedron), A187110 (tetrahedron).
Cf. A126664 (continued fraction), A144535/A144536 (convergents).
Cf. A002194, A010502, A020821, A104956, A152623 (other geometric inequalities).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); Sqrt(3)/2; // G. C. Greubel, Nov 02 2018
  • Maple
    Digits:=100: evalf(sqrt(3)/2); # Wesley Ivan Hurt, Apr 09 2016
  • Mathematica
    RealDigits[Sqrt[3]/2, 10, 200][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)
  • PARI
    default(realprecision, 20080); x=10*(sqrt(3)/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b010527.txt", n, " ", d));  \\ Harry J. Smith, Jun 02 2009
  • PARI
    sqrt(3)/2 \\ Michel Marcus, Apr 10 2016

Formula

Equals cos(30 degrees). - Kausthub Gudipati, Aug 15 2011
Equals A002194/2. - Stanislav Sykora, Nov 30 2013
From Amiram Eldar, Jun 29 2020: (Start)
Equals sin(Pi/3) = cos(Pi/6).
Equals Integral_{x=0..Pi/3} cos(x) dx. (End)
Equals 1/(10*A020832). - Bernard Schott, Sep 29 2022
Equals x^(x^(x^...)) where x = (3/4)^(1/sqrt(3)) (infinite power tower). - Michal Paulovic, Jun 25 2023
Equals 2F1(-1/4,1/4 ; 1/2 ; 3/4) . - R. J. Mathar, Aug 31 2025

Extensions

Last term corrected and more terms added by Harry J. Smith, Jun 02 2009

A020832 Decimal expansion of 1/sqrt(75).

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Multiplied by 10 this is 2/sqrt(3). - Alonso del Arte, Apr 30 2012
2/sqrt(3) is Hermite's constant gamma_2. - Jean-François Alcover, Sep 02 2014, after Steven Finch.
2/sqrt(3) is the Lorentz factor for an object traveling at half the speed of light. - Sean Stroud, May 05 2019

Examples

			0.1154700538379251529...

Links

Crossrefs

Cf. A010153 (continued fraction, but missing the initial 0), A047235.

Programs

Formula

(csc(Pi/3))/10, where csc is the cosecant function. - Alonso del Arte, Apr 30 2012
Product_{n>=1} ((3*n+1)/(3*n+2))^((-1)^n), with offset 1. (see Hu link). - Michel Marcus, Jun 02 2015
From Amiram Eldar, Aug 02 2020: (Start)
2/sqrt(3) = Sum_{k>=0} binomial(2*k,k)/16^k.
2/sqrt(3) = 1 + Sum_{k>=1} (2*k-1)!!/((2*k)!! * 2^(2*k)). (End)
2/sqrt(3) = Product_{k>=1} (1 - (-1)^k/A047235(k)). - Amiram Eldar, Nov 22 2024

A041133 Denominators of continued fraction convergents to sqrt(75).

Original entry on oeis.org

1, 1, 2, 3, 50, 53, 103, 156, 2599, 2755, 5354, 8109, 135098, 143207, 278305, 421512, 7022497, 7444009, 14466506, 21910515, 365034746, 386945261, 751980007, 1138925268, 18974784295, 20113709563, 39088493858, 59202203421, 986323748594, 1045525952015
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A041132, A020832, A010153.

Programs

  • Magma
    I:=[1, 1, 2, 3, 50, 53, 103, 156]; [n le 8 select I[n] else 52*Self(n-4)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013
  • Mathematica
    Denominator/@Convergents[Sqrt[75], 50] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[-(x^2 - x - 1) (x^4 + 3 x^2 + 1)/(x^8 - 52 x^4 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)
LinearRecurrence[{0,0,0,52,0,0,0,-1},{1,1,2,3,50,53,103,156},40] (* Harvey P. Dale, Aug 03 2024 *)

Formula

G.f.: -(x^2-x-1)*(x^4+3*x^2+1) / (x^8-52*x^4+1). - Colin Barker, Nov 13 2013
a(n) = 52*a(n-4) - a(n-8). - Vincenzo Librandi, Dec 11 2013
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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