${H}_{0}=0,{H}_{1}=1,{H}_{2}=1$, for all $n\in \mathbb{N}$ where $n\ge 3$:

Prove for all $n\in \mathbb{N}$,

${H}_{n}={H}_{n-1}+{H}_{n-2}-{H}_{n-3}.$

${H}_{n}=\{\begin{array}{ll}{\displaystyle \frac{n}{2}},& \text{if}n\text{is even}\\ {\displaystyle \frac{n+1}{2}},& \text{if}n\text{is odd}\end{array}$

I don't know how the inductive step $k+1$ in a strong induction would go for piecewise function like this. I think I'll have to show the proposition hold when $k+1$ is even and odd, but I don't know how to continue the proof.