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 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 $p_0 = 1, q_0 = 1, p_{-1} = 1, q_{-1} = 0$.

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 $p_0 = 1, q_0 = 1, p_{-1} = 1, q_{-1} = 0$.

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.

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 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 $p_0 = 1, q_0 = 1, p_{-1} = 1, q_{-1} = 0$.