Non-Integral Value of 3-Göbel Sequence

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Theorem

Consider the $3$-Göbel sequence defined recursively as:

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


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


Proof




Sources