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Fix a positive integer $r\geq 2$. For each integer $k\geq 2$, we have the recursion $$ \left(k + 2\right)\left(k + \frac{1}{3}\right)a_{k} - 2\left(k + 1\right) \left(k - 1 + r\right)a_{k-1} + \left(k - 2 + r\right)\left(k - \frac{1}{3} + r\right)a_{k-2} = 0 $$ Moreover, $a_{0} = 1$ and $a_{1} = r$. I have no experience whatsoever with difference equations. So I was wondering if I could get some advice or suggestions on finding a solution ( assuming it exists ).

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