Three's a crowd Reloaded

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 \(a + b + c\).


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