jmc

To get started, we compute the first ten terms as: $$2007,2008,-4014,2008,2009,-4013,2009,2010,-4012,2010,\dots$$It appears that each term is 1 greater than the number three terms previous. We can demonstrate that this will always occur using the given recurrence relation. We know that $a_n+a_{n+1}+a_{n+2}=n$ and that $a_{n+1}+a_{n+2}+a_{n+3}=n+1$. Subtracting the former from the latter yields $a_{n+3}-a_n=1$, which is the pattern that we observed. Therefore we find that $$a_1=2007,a_4=2008,a_7=2009,\dots ,a_{1000}=2007+333=\boxed{2340}.$$