$a_n = c_1 \lambda_1^n + c_2 \lambda_2^n$ where $\lambda_1, \lambda_2$ are distinct real numbers

$a_n = (c_1 \cos(n\theta) + c_2 \sin(n\theta)) \cdot \rho^n$ for some $\rho$ and $\theta$

$a_n = (c_1 + c_2 n) \lambda^n$ for some repeated root $\lambda$

$a_n = c_1 \lambda^n + c_2 \lambda^{2n}$ for some $\lambda$