Find all $h:\mathbb{N}\rightarrow \mathbb{N}$ such that $$h(h(n)) + h(n+1) = n+2$$
I tried this, but wasn't able to make any progress after a while. So, in vain, I looked at the solution. The solution basically obtained $h(1)=1$ and $h(2)=2$ after some casework, and then claimed that $$h(n) = \lfloor n\alpha \rfloor -n + 1$$ (where $\alpha$ denotes the golden ratio) without explaining at all why one would come at such a conclusion. How did they jump to this conclusion?
(Source: Functional Equations: A Problem Solving Approach, Venkatachala.)