finding a formula for this summation

Question

$$\sum_{i=0}^{(n/2) - 1}{n-2i \choose 2}$$

Where $n$ is an even natural number. For example : If n=8 then I want the summation for this as a formula:

$${8 \choose 2}+{6 \choose 2}+....+{2 \choose 2}$$

I need a formula for this equation to calculate complexity for an algorithm that does a Complete Search: Using recursive backtracking and trying all possible pairings on 2D grid.

Answer 1

Let $n=2k$ for natural number $k$. Then the summation equals :

$$\sum_{i=0}^{k-1}{2k-2i\choose 2}$$ $$=\sum_{i=0}^{k-1}{\frac{(2k-2i)!}{2!(2k-2i-2)!}}$$ $$=\sum_{i=0}^{k-1}{2(k-i)^2}-\sum_{i=0}^{k-1}{(k-i)}$$ $$ =2\sum_{i=1}^{k}{i^2}-\sum_{i=1}^{k}{i}$$

I hope you can take it over from here.