I have been working for quite a while on finding a closed formula for the Legendre Symbol. Inspite of my best efforts I can't come anything better with a formula for the symbol $\left(\dfrac{q}{p}\right)$ of the form $$f(q,p)+\displaystyle\sum_{k=1}^{\frac{p-1}{2}}\left \lfloor \dfrac{kq}{p}\right \rfloor-\left \lfloor \dfrac{k(q-1)}{(p-1)}\right \rfloor$$ For two odd primes $p$ and $q$ with $p>q$. The term $f(p,q)$ has a closed form but I can't find a closed form for the second expression. I have searched in the internet for getting any clue as to how to determine the closed form for this function but the only thing that I found was that $\displaystyle\sum_{k=1}^{\frac{p-1}{2}}\left \lfloor \dfrac{kq}{p}\right \rfloor$ in general has no closed form. But it may be the case that the sums individually may have no closed form (though the second sum has a closed form) but the difference has. 

Also, not the explicit sum but comments regarding its parity will be enough. 

>Notice that the sum can be easily obtained if we can find a closed form for the number of points on the boundary lines or within a triangle with vertices $(0,0)$,$\left(\frac{p-1}{2},\frac{q-1}{2} \right)$ and $\left(\frac{p-1}{2},\frac{q(p-1)}{2p} \right)$.

So, is there any method of obtaining a closed formula for the sum? If not, then can some references be given which inspects this kind of sums? Any help will be appreciated.