Inspired by Pi Han Goh!

$d_n$ will divide $2a_n+2a_{n+1}-(a_{n+1}-a_n)^2=2001$ . Also, $d_{1000}=\boxed{2001}$

If $a_n = n^2 + c$ for some fixed constant $c$ , the answer is always $4c+1$ . To see this, first note that $a_{2c}$ and $a_{2c+1}$ are divisible by $4c+1$ . Then note that $(2n+3)a_n - (2n-1)a_{n+1} = 4c+1$ , so $d_n | (4c+1)$ . So the maximum value of $d_n$ is $4c+1$ , and this maximum is attained.