For positive real numbers $$x$$ and $$y$$, define their special mean to be the average of their arithmetic and geometric means. Find the total number of pairs of integers $$(x,y)$$, with $$x \le y$$, from the set of numbers $$\{1,2,3...,2016\}$$, such that the special mean of $$x$$ and $$y$$ is a perfect square.