Putnam 1990 A1

Let $$T_0 = 2, T_1 = 3, T_2 = 6,$$ and for $n \geq 3$, $$T_n = (n+4)T_{n-1} - 4n T_{n-2} + (4n-8) T_{n-3}.$$ The first few terms are $$2, 3, 6, 14, 40, 152, 784, 5168, 40576.$$ Find, with proof, a formula for $T_n$ of the form $T_n = A_n + B_n$, where $\{A_n\}$ and $\{B_n\}$ are well-known sequences.