A002947 -id:A002947 - cp's OEIS frontend

A005480 Decimal expansion of cube root of 4.

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

Offset: 1

N. J. A. Sloane; entry revised Apr 23 2006

Let h = 4^(1/3). Then (h+1,0) is the x-intercept of the shortest segment from the x-axis through (1,2) to the y-axis; see A197008. - Clark Kimberling, Oct 10 2011

Let h = 4^(1/3). The relative maximum of xy(x+y)=1 is (-1/sqrt(h), h). - Clark Kimberling, Oct 05 2020

			1.587401051968199474751705639272308260391493327899853...

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), p. 173-176.

Harry J. Smith, Table of n, a(n) for n = 1..20000
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. [Annotated scanned copies of pages 175 and 176 only]
Index entries for algebraic numbers, degree 3

Cf. A002947 (continued fraction). - Harry J. Smith, May 07 2009

Cf. A002580 (cube root of 2).

RealDigits[N[4^(1/3), 200]] (* Vladimir Joseph Stephan Orlovsky, May 27 2010 *)

default(realprecision, 20080); x=4^(1/3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b005480.txt", n, " ", d));  \\ Harry J. Smith, May 07 2009, with a correction made May 19 2009

Equals Product_{k>=0} (1 + (-1)^k/(3*k + 1)). - Amiram Eldar, Jul 25 2020
Equals A002580^2. - Michel Marcus, Jan 08 2022
Equals hypergeom([1/3, 1/6], [2/3], 1). - Peter Bala, Mar 02 2022

A002945 Continued fraction for cube root of 2.

Original entry on oeis.org

1, 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, 4, 1, 1, 2, 14, 3, 12, 1, 15, 3, 1, 4, 534, 1, 1, 5, 1, 1, 121, 1, 2, 2, 4, 10, 3, 2, 2, 41, 1, 1, 1, 3, 7, 2, 2, 9, 4, 1, 3, 7, 6, 1, 1, 2, 2, 9, 3, 1, 1, 69, 4, 4, 5, 12, 1, 1, 5, 15, 1, 4

Offset: 0

N. J. A. Sloane

			2^(1/3) = 1.25992104989487316... = 1 + 1/(3 + 1/(1 + 1/(5 + 1/(1 + ...)))).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Harry J. Smith, Table of n, a(n) for n = 0..19999
BCMATH, Continued fraction expansion of the n-th root of a positive rational.
E. Bombieri and A. J. van der Poorten, Continued fractions of algebraic numbers, In: W. Bosma, A. van der Poorten (eds), Computational Algebra and Number Theory. Mathematics and Its Applications, vol. 325.
Ashok Kumar Gupta and Ashok Kumar Mittal, Bifurcating continued fractions, arXiv:math/0002227 [math.GM] (2000).
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134.
S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134. [Annotated scanned copy]
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 13 1973.
Eric Weisstein's World of Mathematics, Delian Constant.
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A002946, A002947, A002948, A002949, A002580 (decimal expansion).

Cf. A002351, A002352 (convergents).

ContinuedFraction(2^(1/3)); // Vincenzo Librandi, Oct 08 2017

N:= 100: # to get a(1) to a(N)
a[1] := 1: p[1] := 1: q[1] := 0: p[2] := 1: q[2] := 1:
for n from 2 to N do
  a[n] := floor((-1)^(n+1)*3*p[n]^2/(q[n]*(p[n]^3-2*q[n]^3)) - q[n-1]/q[n]);
  p[n+1] := a[n]*p[n] + p[n-1];
  q[n+1] := a[n]*q[n] + q[n-1];
od:
seq(a[i],i=1..N); # Robert Israel, Jul 30 2014

ContinuedFraction[Power[2, (3)^-1],70] (* Harvey P. Dale, Sep 29 2011 *)

allocatemem(932245000); default(realprecision, 21000); x=contfrac(2^(1/3)); for (n=1, 20000, write("b002945.txt", n-1, " ", x[n])); \\ Harry J. Smith, May 08 2009

From Robert Israel, Jul 30 2014: (Start)
Bombieri/van der Poorten give a complicated formula:
a(n) = floor((-1)^(n+1)*3*p(n)^2/(q(n)*(p(n)^3-2*q(n)^3)) - q(n-1)/q(n)),
p(n+1) = a(n)*p(n) + p(n-1),
q(n+1) = a(n)*q(n) + q(n-1),
with a(1) = 1, p(1) = 1, q(1) = 0, p(2) = 1, q(2) = 1. (End)

BCMATH link from Keith R Matthews (keithmatt(AT)gmail.com), Jun 04 2006

Offset changed by Andrew Howroyd, Jul 04 2024

cp's OEIS Frontend

A005480 Decimal expansion of cube root of 4.

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