

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
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OFFSET

1,2


REFERENCES

S. Lang and H. Trotter, Continued fractions for some algebraic numbers, J. Reine Angew. Math. 255 (1972), 112134.
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 nth 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)^32*q(n)^3))  q(n1)/q(n)),
p(n+1) = a(n)*p(n) + p(n1),
q(n+1) = a(n)*q(n) + q(n1),
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]^32*q[n]^3))  q[n1]/q[n]);
p[n+1] := a[n]*p[n] + p[n1];
q[n+1] := a[n]*q[n] + q[n1];
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



