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%I M1354 N0521 #97 May 03 2024 15:48:44
%S 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,
%T 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,
%U 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
%N Decimal expansion of cube root of 2.
%C 2^(1/3) is Hermite's constant gamma_3. - _Jean-François Alcover_, Sep 02 2014, after Steven Finch.
%C 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
%C Length of an edge of a cube with volume 2. - _Jared Kish_, Oct 16 2014
%C 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
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D 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.
%H Harry J. Smith, <a href="/A002580/b002580.txt">Table of n, a(n) for n = 1..20000</a>
%H Laszlo C. Bardos, <a href="https://www.cutoutfoldup.com/409-double-a-cube.php">Double a Cube</a>, CutOutFoldUp.
%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 62.
%H Wolfdieter Lang, <a href="http://arxiv.org/abs/1409.4799">Notes on Some Geometric and Algebraic Problems solved by Origami</a>, arXiv:1409.4799 [math.MG], 2014.
%H Liz Newton, <a href="http://plus.maths.org/content/os/issue53/features/newton/index">The power of origami</a>.
%H Raul Prisacariu, <a href="https://web.archive.org/web/20231208023740/https://www.raulprisacariu.com/math/lills-method-and-the-philo-line-for-right-angles/">Lill's method and the Philo Line for Right Angles</a>.
%H Simon Plouffe, Plouffe's Inverter, <a href="http://www.plouffe.fr/simon/constants/cuberoot2.txt">The cube root of 2 to 20000 digits</a>
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap16.html">2**(1/3) to 2000 places</a>
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/gendev/125992.html">Generalized expansion of real constants</a>
%H H. S. Uhler, <a href="/A002580/a002580.pdf">Many-figure approximations for cubed root of 2, cubed root of 3, cubed root of 4, and cubed root of 9 with chi2 data</a>, Scripta Math. 18, (1952). 173-176. [Annotated scanned copies of pages 175 and 176 only]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DelianConstant.html">Delian Constant</a>
%H <a href="/index/Al#algebraic_03">Index entries for algebraic numbers, degree 3</a>
%F (-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
%F 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
%F Equals Product_{k>=0} (1 + (-1)^k/(3*k + 2)). - _Amiram Eldar_, Jul 25 2020
%F From _Peter Bala_, Mar 01 2022: (Start)
%F 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.
%F 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).
%F Equals hypergeom([-2/3, -1/6], [1/2], 1).
%F Equals hypergeom([2/3, 1/6], [4/3], 1). (End)
%e 1.2599210498948731647672106072782283505702514...
%p Digits:=100: evalf(2^(1/3)); # _Wesley Ivan Hurt_, Nov 12 2015
%t RealDigits[N[2^(1/3), 5!]] (* _Vladimir Joseph Stephan Orlovsky_, Sep 04 2008 *)
%o (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
%o (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
%Y Cf. A002945 (continued fraction), A270714 (reciprocal), A253583.
%Y Cf. A246644.
%Y Cf. A002581, A005480, A005481, A005482, A005486, A010581, A010582, A092039, A092041, A139340.
%K nonn,easy,cons,changed
%O 1,2
%A _N. J. A. Sloane_
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