A002945 Continued fraction for cube root of 2. (Formerly M2220)

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

Offset

1,2

References

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

Links

Harry J. Smith, Table of n, a(n) for n = 1..20000

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

Formula

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)

Example

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

Maple

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

Mathematica

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

Prog

(PARI) allocatemem(932245000); default(realprecision, 21000); x=contfrac(2^(1/3)); for (n=1, 20000, write("b002945.txt", n, " ", x[n])); \\ Harry J. Smith, May 08 2009
(Magma) ContinuedFraction(2^(1/3)); // Vincenzo Librandi, Oct 08 2017

Crossrefs

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

Sequence in context: A187367 A307410 A305444 * A171232 A093423 A326454

Adjacent sequences: A002942 A002943 A002944 * A002946 A002947 A002948

Keyword

cofr,nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved