Non-Integral Value of 3-Göbel Sequence

Theorem
Consider the $3$-Göbel sequence defined recursively as:
 * $x_n = \begin {cases}

1 & : n = 0 \\ \displaystyle \paren {1 + \sum_{k \mathop = 0}^{n - 1} {x_k}^3} / n & : n > 0 \end {cases}$

The smallest $n$ such that $x_n$ is not an integer is $88$.