A002945 Continued fraction for cube root of 2.
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
Examples
2^(1/3) = 1.25992104989487316... = 1 + 1/(3 + 1/(1 + 1/(5 + 1/(1 + ...)))).
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 = 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
Crossrefs
Programs
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Magma
ContinuedFraction(2^(1/3)); // Vincenzo Librandi, Oct 08 2017
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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
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Mathematica
ContinuedFraction[Power[2, (3)^-1],70] (* Harvey P. Dale, Sep 29 2011 *)
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PARI
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
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)
Extensions
BCMATH link from Keith R Matthews (keithmatt(AT)gmail.com), Jun 04 2006
Offset changed by Andrew Howroyd, Jul 04 2024