Recurring Styles #9 - Integers for a Condition!

\[\large{(n+1)(n-2)x_{n+1} = n(n^2 - n - 1)x_n - (n-1)^3 x_{n-1}}\]

Let \((x_n)_{n \geq 2}\) be a sequence such that \(x_2 = 1, x_3 = 1\), and for all \(n \geq 3\), the above recurrence relation satisfies.

For which all values of \(n, \ n \geq 4\) is every \(x_n\) an integer?

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