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.

  • 3
    $\begingroup$ The recurrence is a quadratic map, and those don't have "nice" solutions in general. This one reduces to $\,b_{n+1} = 2 b_n^2 - 3\,$ with $\,a_n = b_n - \frac{3}{2}\,$, which doesn't look much better. $\endgroup$ – dxiv Jan 25 '18 at 5:23

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