Largest Roots of Cubic Polynomials
Let $a_n$ be the largest real root of a polynomial $g(x) = x^3 - 2^n \cdot x^2 + n$.
For example, $a_2 = 3.86619826\cdots$
Find the last eight digits of $\sum \limits_{i = 1}^{30} \lfloor a_i^{987654321} \rfloor$.
Note: $\lfloor a \rfloor$ represents the floor function.