Find an explicit formula for the recursive formula: $$a_{n+1} = 2a_n\left(a_n + 3\right); a_0 = 4$$
The first few terms in the sequence go like this: $4, 56, 6608, \dots$
After $a_2$ the sequence begins increasing at a very strong rate.
Normally how we were taught to find an explicit formula, we start by defining the first few terms of the sequence in terms of the initial term, $a_0$, and then look for patterns to generalize a formula for the $n$th term.
For this example, we have $a_1 = 2(a_0)^2+6(a_0)$, but it only got worse when trying to find $a_2$.
$a_2 = 8(a_0)^4 + 48(a_0)^3+84(a_0)^2 + 36(a_0)$
I feel as if this isn't the most efficient method to find the explicit formula, and I imagine $a_3$ would only be a "messier" polynomial and won't help me give me any sort of clue as to what the explicit formula may be.
Is there another way to tackle this problem?
Note: although I was never taught this method, I hear generating functions may be able to be used for problems such as these.