Let $$A_{n}$$ be the number of ways we can arrange $$n$$ identical black balls and $$n$$ identical red balls in a line, such that no more than 2 balls of the same color ever appear in a row.

If the limit of $\lim_{n\to\infty} \dfrac {A_{n+1} - A_{n}} {A_{n}}= \dfrac {a + \sqrt b}{c},$ where $$a, b,$$ and $$c$$ are positive integers with $$a$$ and $$c$$ coprime, submit your answer as $$a + b + c$$.

×