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A002945 Continued fraction for cube root of 2.
(Formerly M2220)
9

%I M2220

%S 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,

%T 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,

%U 7,6

%N Continued fraction for cube root of 2.

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

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

%H Harry J. Smith, <a href="/A002945/b002945.txt">Table of n, a(n) for n = 1..20000</a>

%H BCMATH, <a href="http://www.numbertheory.org/php/cfrac_nthroot.html">Continued fraction expansion of the n-th root of a positive rational</a>

%H E. Bombieri and A. J. van der Poorten, <a href="http://maths.mq.edu.au/~alf/www-centre/alfpapers/a113.pdf">Continued fractions of algebraic numbers</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DelianConstant.html">Delian Constant</a>

%H G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">Contfrac</a>

%H <a href="/index/Con#confC">Index entries for continued fractions for constants</a>

%F Bombieri/van der Poorten give a complicated formula:

%F a(n) = floor((-1)^(n+1)*3*p(n)^2/(q(n)*(p(n)^3-2*q(n)^3)) - q(n-1)/q(n)),

%F p(n+1) = a(n)*p(n) + p(n-1),

%F q(n+1) = a(n)*q(n) + q(n-1),

%F with a(1) = 1, p(1) = 1, q(1) = 0, p(2) = 1, q(2) = 1. - _Robert Israel_, Jul 30 2014

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

%p N:= 100: # to get a(1) to a(N)

%p a[1] := 1: p[1] := 1: q[1] := 0: p[2] := 1: q[2] := 1:

%p for n from 2 to N do

%p 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 p[n+1] := a[n]*p[n] + p[n-1];

%p q[n+1] := a[n]*q[n] + q[n-1];

%p od:

%p seq(a[i],i=1..N); # _Robert Israel_, Jul 30 2014

%t ContinuedFraction[Power[2, (3)^-1],70] (* _Harvey P. Dale_, Sep 29 2011 *)

%o (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]

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

%K cofr,nonn

%O 1,2

%A _N. J. A. Sloane_.

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

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Last modified December 22 21:10 EST 2014. Contains 252372 sequences.