I'm having trouble simplifying the expression for how many sets I can possibly have.
It's a very specific problem for which the specifics don't actually matter, but for $q$, some power of $2$ greater than $4$, I have a set of $q - 3$ elements. I am finding all subsets which contain at least two elements and at most half ($q / 2$, which would then also be a power of $2$) of the elements. I know that the total number of subsets which satisfy these conditions is (sorry my TeX may be awful),
$\sum\limits_{i=2}^{\frac{q}{2}} \binom{q-3}{i}= \sum\limits_{i=0}^{\frac{q}{2}} \binom{q-3}{i}- (q - 3 + 1)$
but I'm having a tough time finding a closed-form expression for this summation. I'm probably missing something, but it has stumped me this time.