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 1-4 of 4 results.

A002580 Decimal expansion of cube root of 2.

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

Comments

2^(1/3) is Hermite's constant gamma_3. - Jean-François Alcover, Sep 02 2014, after Steven Finch.
For doubling the cube using origami and a standard geometric construction employing two right angles see the W. Lang link, Application 2, p. 14, and the references given there. See also the L. Newton link. - Wolfdieter Lang, Sep 02 2014
Length of an edge of a cube with volume 2. - Jared Kish, Oct 16 2014
For any positive real c, the mappings R(x)=(c*x)^(1/4) and S(x)=sqrt(c/x) have the same unique attractor c^(1/3), to which their iterated applications converge from any complex plane point. The present case is obtained setting c=2. It is noteworthy that in this way one can evaluate cube roots using only square roots. The CROSSREFS list some other cases of cube roots to which this comment might apply. - Stanislav Sykora, Nov 11 2015
The cube root of any positive number can be connected to the Philo lines (or Philon lines) for a 90-degree angle. If the equation x^3-2 is represented using Lill's method, it can be shown that the path of the root 2^(1/3) creates the shortest segment (Philo line) from the x axis through (1,2) to the y axis. For more details see the article "Lill's method and the Philo Line for Right Angles" linked below. - Raul Prisacariu, Apr 06 2024

Examples

			1.2599210498948731647672106072782283505702514...

References

  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 192-193.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.4 Irrational Numbers and §12.3 Euclidean Construction, pp. 84, 421.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • Horace S. Uhler, Many-figure approximations for cubed root of 2, cubed root of 3, cubed root of 4, and cubed root of 9 with chi2 data. Scripta Math. 18, (1952). 173-176.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, pp. 33-34.

Links

Crossrefs

Cf. A002945 (continued fraction), A270714 (reciprocal), A253583.
Cf. A246644.
Cf. A002581, A005480, A005481, A005482, A005486, A010581, A010582, A092039, A092041, A139340.

Programs

  • Maple
    Digits:=100: evalf(2^(1/3)); # Wesley Ivan Hurt, Nov 12 2015
  • Mathematica
    RealDigits[N[2^(1/3), 5!]] (* Vladimir Joseph Stephan Orlovsky, Sep 04 2008 *)
  • PARI
    default(realprecision, 20080); x=2^(1/3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002580.txt", n, " ", d));  \\ Harry J. Smith, May 07 2009
  • PARI
    default(realprecision, 100); x= 2^(1/3); for(n=1, 100, d=floor(x); x=(x-d)*10; print1(d, ", "))  \\ Altug Alkan, Nov 14 2015

Formula

(-2^(1/3) - 2^(1/3) * sqrt(-3))^3 = (-2^(1/3) + 2^(1/3) * sqrt(-3))^3 = 16. - Alonso del Arte, Jan 04 2015
Set c=2 in the identities c^(1/3) = sqrt(c/sqrt(c/sqrt(c/...))) = sqrt(sqrt(c*sqrt(sqrt(c*sqrt(sqrt(...)))))). - Stanislav Sykora, Nov 11 2015
Equals Product_{k>=0} (1 + (-1)^k/(3*k + 2)). - Amiram Eldar, Jul 25 2020
From Peter Bala, Mar 01 2022: (Start)
Equals Sum_{n >= 0} (1/(3*n+1) - 1/(3*n-2))*binomial(1/3,n) = (3/2)* hypergeom([-1/3, -2/3], [4/3], -1). Cf. A290570.
Equals 4/3 - 4*Sum_{n >= 1} binomial(1/3,2*n+1)/(6*n-1) = (4/3)*hypergeom ([1/2, -1/6], [3/2], 1).
Equals hypergeom([-2/3, -1/6], [1/2], 1).
Equals hypergeom([2/3, 1/6], [4/3], 1). (End)

A002358 Numerators of convergents to cube root of 5.

Original entry on oeis.org

1, 2, 5, 12, 53, 171, 566, 737, 4251, 4988, 9239, 41944, 428679, 7329487, 7758166, 115943811, 123701977, 239645788, 731522646953, 731762292741, 1463284939694, 2195047232435, 3658332172129, 5853379404564, 9511711576693, 24876802557950, 59265316692593
Offset: 0

Views

Author

N. J. A. Sloane, Herman P. Robinson

Keywords

References

  • D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 67.
  • P. Seeling, Verwandlung der irrationalen Groesse ... in einen Kettenbruch, Archiv. Math. Phys., 46 (1866), 80-120.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002357 (denominators), A005481.

Programs

  • Mathematica
    Numerator[Convergents[5^(1/3), 30]] (* Vincenzo Librandi, Sep 08 2013 *)
  • PARI
    a(n)= contfracpnqn(contfrac(5^(1/3), n))[1, 1]; \\ Michel Marcus, Sep 07 2013

Extensions

More terms from Vincenzo Librandi, Sep 08 2013

A002948 Continued fraction for cube root of 5.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 3, 1, 5, 1, 1, 4, 10, 17, 1, 14, 1, 1, 3052, 1, 1, 1, 1, 1, 1, 2, 2, 1, 3, 2, 1, 13, 5, 1, 1, 1, 13, 2, 41, 1, 4, 12, 1, 5, 2, 7, 1, 1, 3, 33, 2, 1, 1, 1, 1, 1, 1, 3, 2, 2, 1, 15, 12, 8, 10, 48, 1, 2, 1, 1, 3, 4, 1, 474, 1, 13, 2, 4, 1, 1, 49
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			5^(1/3) = 1.70997594667669698... = 1 + 1/(1 + 1/(2 + 1/(2 + 1/(4 + ...)))). - _Harry J. Smith_, May 08 2009

References

  • H. P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A005481 (decimal expansion).
Cf. A002357, A002358 (convergents).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); ContinuedFraction(5^(1/3)); // G. C. Greubel, Nov 02 2018
  • Maple
    with(numtheory): cfrac(5^(1/3),80,'quotients'); # Muniru A Asiru, Nov 02 2018
  • Mathematica
    ContinuedFraction[5^(1/3), 100] (* G. C. Greubel, Nov 02 2018 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 21000); x=contfrac(5^(1/3)); for (n=1, 20000, write("b002948.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 08 2009

Extensions

Offset changed by Andrew Howroyd, Jul 04 2024

A002357 Denominators of convergents to cube root of 5.

Original entry on oeis.org

1, 1, 3, 7, 31, 100, 331, 431, 2486, 2917, 5403, 24529, 250693, 4286310, 4537003, 67804352, 72341355, 140145707, 427797039119, 427937184826, 855734223945, 1283671408771, 2139405632716, 3423077041487, 5562482674203, 14548042389893
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			1 + 1/(1 + 1/(2 + 1/(2 + 1/4))) = 53/31 so a(5) = 31.

References

  • D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 67.
  • P. Seeling, Verwandlung der irrationalen Groesse ... in einen Kettenbruch, Archiv. Math. Phys., 46 (1866), 80-120.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Crossrefs

Cf. A002358 (numerators), A005481.

Programs

  • PARI
    a(n)=if(n<1,0,contfracpnqn(contfrac(5^(1/3),n))[2,1])

Extensions

More terms from Herman P. Robinson
Offset changed by Andrew Howroyd, Jul 04 2024
Showing 1-4 of 4 results.

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