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



Are there any patterns in simple continued fraction expansions of algebraic real numbers?
For a "formula" for the continued fraction of algebraic numbers, in particular $2^{1/3}$, see [Bombieri and van der Poorten](http://maths.mq.edu.au/~alf/www-centre/alfpapers/a113.pdf). 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
$p\_0 = 1, q\_0 = 1, p\_{-1} = 1, q\_{-1} = 0$.


