If I have a recurrence relation, such as $h_n = h_{n-1} + 2h_{n-2}$, is there a rigid method to find a closed formula for $h_n$?
As of right now, I just solve for the first few terms $h_0, h_1, h_2, h_3$, etc. until I notice a pattern and try to brute force an answer, but some patterns are not easily discerned by such a method (such as this one).
For reference this particular sequence is: $1, 3, 5, 11, 21, 43, \dots$
I was given that $h_0 = 1$, and $h_1 = 3$.
I figured out that the answer is $h_n = \dfrac{2^{n+2}-(-1)^{n+2}}{3}$ after a lot of trial and error. Is there a better way to find this?