Non-Integral Value of Göbel's Sequence

Theorem
Consider Göbel's sequence defined recursively as:
 * $x_n = \begin{cases}

1 & : n = 0 \\ \ds \paren {1 + \sum_{k \mathop = 0}^{n - 1} {x_k}^2} / n & : n > 0 \end{cases}$

The smallest $n$ such that $x_n$ is not an integer is $43$.