Start once, second half is easier

Let a recurrence relation be defined as follows: \[a_1=1;\ a_2=3;\ a_{n+2}=(n+3)a_{n+1}-(n+2)a_{n}\ \forall\ n\in N\]

Calculate the sum of all values of \(x\) such that \(a_{x}=x^2\).

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