Holiday Contest 2025 - Team Round - Problem 16

Let a_1, a_2, \ldots a_n be a sequence defined such that a_1 = \tfrac{1}{6} and a_{n-1} = a_{n-2} + \frac{1}{n^3-n}. For how many of the first 100 terms of the sequence is the numerator of a_i even when it is fully simplified?