Drawing Points and Friendliness

A point $(x_1,y_1)$ is drawn in the positive coordinate plane with $x_1 > y_1$ . Then, an infinite sequence of points are drawn, with the following rule: $\left\{\begin{array}{l}x_{n+1}=x_n+y_n\\ y_{n+1}=x_n-y_n\end{array}\right.$

Suppose a starting point $(x_1,y_1)$ is called friendly to a coordinate $(a,b)$ if a point is drawn on $(a,b)$ sometime in the sequence; that is, $(a,b)=(x_k,y_k)$ for some positive integer $k$ .

Let $F(a,b)$ be a function that counts the number of starting points $(x_1,y_1)$ that are friendly to $(a,b)$ .

The number of ordered pairs $(p,q)$ satisfying $F(p,q)=2014$ and $p+q \le 2^{2^{10}}$ can be expressed as $a^b$ where $a,b$ are positive integers and $a$ is minimized. Find the value of $a+b$ .

Details and Assumptions

As much as this looks like a Computer Science problem, it is 100% doable with only a pencil and a paper.