<s>[INST] Find closed form for $a_n=2\frac{n-1}{n}a_{n-1}-2\frac{n-2}{n}a_{n-2}$ for all $n \ge 3$ Find a closed form for:
$a_n=2\frac{n-1}{n}a_{n-1}-2\frac{n-2}{n}a_{n-2}$ for all $n \ge 3$
Initial values: $a_1=3, a_2=1$
I'm struggling a bit with the algebra as the solution contains complex number. Any help would be appreciated.
 [/INST] Set $u_n = na_n $ then your relation becomes 
$$u_n = 2u_{n-1}-2u_{n-2}$$
Whose characteritic equation is $$x^2-2x+2=0 \implies x=\frac{2\pm 2i}{2}$$
And hence $$ u_n =A\left(\frac{2-2i}{2}\right)^{n-1} +B\left(\frac{2+2i}{2}\right)^{n-1}$$

that is $$\color{red}{a_n =\frac{A}{n}\left(\frac{2-2i}{2}\right)^{n-1} +\frac{B}{n}\left(\frac{2+2i}{2}\right)^{n-1}}$$
  Now can you fine A and B?

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