Consider the sequence defined by $$ \begin{cases} r_0=1\\ r_1=3\\ r_n=6r_{n-1}-9r_{n-2} & \text{if }n\ge 2 \end{cases} .$$ Find a closed form for $r_n$.
Your response should be a formula in terms of $n$, and should not contain terms such as $r_n,$ $r_{n-1},$ and so on. Do not include $``r_n=\text{''}$ in your response.
I realize that the characteristic equation of the recurrence is $c^2-6c+9$ and that can be factored into $(c-3)(c-3)$. So, I then have $$r_{0} = \lambda_1+\lambda_2$$ and $$r_1 = 3\lambda_1+3\lambda_2.$$ But shouldn't this have infinite solutions?
Hint. This is a constant-recursive sequence in which the characteristic polynomial has a double root equal to $3$. In this case, the general term has the form $c_n = (\lambda_1 + n \lambda_2)3^n$.