BALKAN-SL 2008 N6

Let $(x_n)$ , $n=1,2, \ldots$ be a sequence defined by $x_1=2008$ and \begin{align*} x_1 +x_2 + \ldots + x_{n-1} = \left( n^2-1 \right) x_n ~ ~ ~ \forall n \geq 2 \end{align*} Let the sequence $a_n=x_n + \frac{1}{n} S_n$ , $n=1,2,3, \ldots$ where $S_n$ $=$ $x_1+x_2 +\ldots +x_n$ . Determine the values of $n$ for which the terms of the sequence $a_n$ are perfect squares of an integer.