Putnam 2013 B1

For positive integers $n$, let the numbers $c(n)$ be determined by the rules $c(1) = 1$, $c(2n) = c(n)$, and $c(2n+1) = (-1)^n c(n)$. Find the value of $$\sum_{n=1}^{2013} c(n) c(n+2).$$