A002580 Decimal expansion of cube root of 2.

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• cons

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

• Harry J. Smith, Table of n, a(n) for n = 1..20000
• Laszlo C. Bardos, Double a Cube, CutOutFoldUp.
• Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 62.
• Wolfdieter Lang, Notes on Some Geometric and Algebraic Problems solved by Origami, arXiv:1409.4799 [math.MG], 2014.
• Liz Newton, The power of origami.
• Raul Prisacariu, Lill's method and the Philo Line for Right Angles.
• Simon Plouffe, Plouffe's Inverter, The cube root of 2 to 20000 digits.
• Simon Plouffe, 2**(1/3) to 2000 places.
• Simon Plouffe, Generalized expansion of real constants.
• H. 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. [Annotated scanned copies of pages 175 and 176 only]
• Eric Weisstein's World of Mathematics, Delian Constant.
• Index entries for algebraic numbers, degree 3.

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)