16. The Reckoning of a Recurrence

For positive integers $n$, define the sequence $c(n)$ by

$$c(1)=1,$$ $$c(2n)=c(n),$$

and

$$c(2n+1)=(-1)^n c(n).$$

Find the value of

$$\sum_{n=1}^{2027} c(n)c(n+6).$$