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 A002945 Continued fraction for cube root of 2. (Formerly M2220) 16
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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 = 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 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-1, " ", 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). Cf. A002351, A002352 (convergents). 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 Offset changed by Andrew Howroyd, Jul 04 2024 STATUS approved

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Last modified August 10 05:56 EDT 2024. Contains 375044 sequences. (Running on oeis4.)