Find, with proof, a formula for T n , of the form T = A n +B

Find, with proof, a formula for T_n, of the form T₁ = A_n +B_n where {A_n} and {B_n} are well-known sequences.
Let T₀ = 2, T₁ = 3, T₂ = 6, and for n ≥ 3,

T₁ = (n + 4)T_n−1 − 4nT_n-2 + (4n − 8)T_n-3.

The first few terms are

2, 3, 6, 14, 40, 152, 784, 5168, 40,576