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A002945 Continued fraction for cube root of 2.
(Formerly M2220)
9
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112-134.

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

Eric Weisstein's World of Mathematics, Delian Constant

G. Xiao, Contfrac

Index entries for continued fractions for constants

FORMULA

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. - Robert Israel, Jul 30 2014

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]

CROSSREFS

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

Sequence in context: A010286 A176801 A187367 * A171232 A093423 A227507

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

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Last modified September 30 19:57 EDT 2014. Contains 247475 sequences.