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$x + y + z = 0$
$x + y = - z$ $x^3+y^3+z^3=(x+y) \cdot (x^2-xy+y^2)+z^3= -z \cdot (x^2-xy+y^2) + z^3=xyz-z \cdot (x^2+y^2-z^2) = xyz - z \cdot (x^2+y^2-(x+y)^2) = xyz - z \cdot (x^2+y^2-x^2-2xy-y^2) = xyz +2xyz=3xyz$ $a_n=- \frac {2 \sqrt {5}+6}{\sqrt {5}-5} \cdot (\frac {1+\sqrt {5}} {2})^{n-1}- \frac {2 \sqrt {5}-6}{\sqrt {5}+5} \cdot (\frac {1 - \sqrt {5}} {2})^{n-1}-3$

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