Is it possible to obtain a solution for $a_n$ for the following recursion relation?
$$a_n = -\frac{1}{\epsilon}(n+1)a_{n+1}+R\left(\frac{n+2}{n+1}\right)a_{n+2}+\frac{R}{\epsilon}\left(\frac{(n+3)(n+2)}{n+1}\right)a_{n+3}$$
If not, any other methods and hints on how to get rid of the recursion will be appreciated.
Maple gives:
$$a_n=\frac{(-\epsilon)^n}{\Gamma(n+1)}\left(a_0+\sum_{i=0}^{n-1}t\right)$$
$$t=-\frac{1}{\pi\epsilon}\frac{4}{\epsilon^2R}^\frac{i}{2}(a_0\epsilon+a_1)\Gamma\left(\frac{i+1}{2}\right)^2\ \ \ i\colon even$$
$$t=\frac{4}{\epsilon^2R}^\frac{i-1}{2}\Gamma\left(\frac{i+1}{2}\right)^2\left(\frac{2a_2}{\epsilon^2}+\frac{a_1}{\epsilon}\right)\ \ \ i\colon odd$$