$a_n = 4a_{n-1} - 4a_{n-2}$ (homogeneous only)

$a_n = 4a_{n-1} - 4a_{n-2} + 2$ (constant non-homogeneous term)

$a_n = 4a_{n-1} - 4a_{n-2} + 2^n$ (exponential non-homogeneous term)

$a_n = 2a_{n-1} - a_{n-2} + n$ (polynomial non-homogeneous term)