Interesting Recursive Sequence

Question

I was messing with sequences defined recursively in terms of the previous two terms and I came across the sequence $$P_n=4P_{n-2}-P_{n-1}$$ I have been trying to find two seeds $P_1$ and $P_2$ so that no value of $P_n$ is negative, and I haven't been able to do it. I have not yet tried writing an explicit formula in terms of the seeds, but it will be messy and inelegant if I do. Is there another way to find two such seeds or prove their nonexistence? Thanks!

Answer 1

The general solution of this recursion is $a r_1^n + b r_2^n$ where $r_1 = (-1-\sqrt{17})/2$ and $r_2 = (-1+\sqrt{17})/2$ are the roots of the polynomial $x^2 + x -4$. Since $r_1 < 0 < r_2$ and $|r_1|>r_2$, the only way to have all terms positive is to have $a=0$ and $b > 0$. Thus you want $P_2 = P_1 r_2$ with $P_1 > 0$.