Divisibility of a recursive sequence.

Let \(a_{1}=1\) and \(a_{2}=3\) and for \(n > 2\) \[a_{n}=(n+1)a_{n-1} - na_{n-2}\] For how many positive integers \(n \leq 1000\) is \(a_{n}\) divisible by \(11\)?

Details and Assumptions

Computation devices are not necessary.

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