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356. Largest roots of cubic polynomials

Let a_n be the largest real root of a polynomial g(x) = x³ - 2ⁿ·x² + n.
For example, a₂ = 3.86619826...

Find the last eight digits of $\sum \limits_{i = 1}^{30} {\left \lfloor a_i^{987654321} \right \rfloor}$.

Note: $\lfloor a \rfloor$ represents the floor function.