$F(0)= 0$ and $F(1) = 1$ are predefined; $F(n)$ references the $n^{th}$ Fibonacci number. $n(h)$ is the minimal number of nodes needed to construct a AVL binary tree of height $h$. The theory shouldn't matter too much here I only have to prove the equation by complete induction. I am already through the induction hypothesis and tried two different values $> 2$ which worked fine as expected.
This is what I did: After swapping $h$ with $k+1$ I got: $1 + n(h) + n(h-1) = F(h+3) - 1$
I then splitted $F(h+3) - 1$ into it's components $F(h+2) + F(h+1) - 1$.
Then $F(h+2) - 1$ could be replaced by $1 + n(h-1) + n(h-2)$ following the hypothesis.
This gets: $1 + n(h) + n(h-1) = 1 + n(h-1) + n(h-2) + F(h+1)$ and this is where I am stuck as I could split the Fibonacci numbers as long as I want without getting any further. I would really appreciate some advice from you guys :/