Solve a recurrence relation with two recursion calls using the iteration method

Question

I can't figure out how to solve this recurrence relation using the iteration method:

$$T(n) = \begin{cases} 0, & \text{if $n=0$} \\ 1, & \text{if $n=1$} \\ 3T(n-1)+ 4T(n-2), & \text{if $n >1$} \end{cases}$$

Answer 1

Hint: Let $S(n) = T(n+1) + T(n)$. We have $S(0) = 1$, and for $n > 0$, $$ S(n) = T(n+1) + T(n) = [3T(n) + 4T(n-1)] + T(n) = 4(T(n) + T(n-1)) = 4S(n-1). $$ Can you take it from here?