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, $a_{2} = 3.86619826 \dots$

Find the last eight digits of $\sum_{i = 1}^{30} ⌊ a_{i}^{987654321} ⌋$ .

Note: $⌊ a ⌋$ represents the floor function.

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.