The Hockey-stick identity is
$$\sum^n_{i=r}{i\choose r}={n+1\choose r+1} \qquad \text{ for } n,r\in\mathbb{N}, \quad n>r$$
I am trying to determine the values of the following
$$\sum_{\substack{i=r \\ i\,\text{even}}}^n{i\choose r}\;\text{and}\sum_{\substack{i=r \\ i\,\text{odd}}}^n{i\choose r}$$
EDIT WITH PARTIAL ANSWERS: The cases $r=0,1$ are clear, and I have been able to prove (i.e. beyond computer guesswork) that for $r=2,3,4$ and $n=2(k-1)$ the even sums are $$\sum_{i=2}^{k}{2(i-1)\choose 2}={k\choose 2}+4{k\choose 3}$$ $$\sum_{i=3}^{k}{2(i-1)\choose 3}=4{k\choose 3}+8{k\choose 4}$$ $$\sum_{i=3}^{k}{2(i-1)\choose 4}={k\choose 3}+12{k\choose 4}+16{k\choose 5}$$ respectively. However, I am still at a loss regarding the general case.