Are there any patterns in simple continued fraction expansions of algebraic real numbers? As we know there are patterns in simple continued fraction expansion of quadratic algebraic numbers,are  there any patterns  in  simple continued fraction expansions of  other algebraic real numbers?Or any law in them?or is there any universal algorithm to compute the integer sequence in simple continued fraction expansions,with the equation whose one solution is the real algebraic number
 A: Continued fractions for complex numbers can be studied.  However, there is no "canonical" choice of which possible continued fraction to use.  And, for that matter, why use the Gaussian integers $\mathbb Z[\sqrt{-1}\;]$ and not some other chioce?
See HERE for an article:
Convergence of Complex Continued Fractions
by John Marafino (James Madison University) and Timothy J. McDevitt (James Madison University)
This article originally appeared in:
Mathematics Magazine
June, 1995  
Convergence of a complex continued fraction can be analyzed using analysis, algebra, number theory, topology or complex analysis.
A: For a "formula" for the continued fraction of algebraic numbers, in particular $2^{1/3}$, see Bombieri and van der Poorten.  It's just not a simple pattern.
EDIT: Actually there's an error in the formula in the middle of page 152 there:
it should be
$$ \pmatrix{p_{h+1} & q_{h+1}\cr p_h & q_h\cr} = \pmatrix{c_{h+1} & 1\cr
1 & 0\cr} \pmatrix{p_h & q_h\cr p_{h-1} & q_{h-1}\cr} $$
That is, the recurrence for the continued fraction $1 + \frac{1}{c_1 + \frac{1}{c_2 + \ldots}}$ of $2^{1/3}$ is
$$ \eqalign{  c_{h+1} &= \left\lfloor {\frac { 3 \;\left( -1 \right) ^{h+1}{p_{{h}}}^{2}}{q_{{h}
} \left( {p_{{h}}}^{3}-2\,{q_{{h}}}^{3} \right) }}-{\frac {q_{{h-1}}}{
q_{{h}}}}\right\rfloor\cr
p_{h+1} &= c_{h+1} p_h + p_{h-1}\cr
q_{h+1} &= c_{h+1} q_h + q_{h-1}\cr}
$$
with initial values
$p_0 = 1, q_0 = 1, p_{-1} = 1, q_{-1} = 0$.
