2020-13 An integral sequence

Let $$a_n$$ be a sequence defined recursively by $$a_0 = a_1 = \dots = a_5 = 1$$ and
$a_n = \frac{a_{n-1} a_{n-5} + a_{n-2} a_{n-4} + a_{n-3}^2}{a_{n-6}}$
for $$n \geq 6$$. Prove or disprove that every $$a_n$$ is an integer.

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