Like many problems here they can be solved analytically, and this always adds another layer to the problem for the committed.

In my post above I showed that the sum of four corners for an nxn grid is given by 4n2-6n+6.

So we need to add that expression from n=3 to n=1001 in steps of 2.

The sum of the first k odd squares is k(4k2-1)/3.
The sum of the first k odd numbers is k2.

So we get S=4k(4k2-1)/3-6k2+6k.

Substituting in k=501 (1001 is the 501st odd), we get S=669171004, but this sum includes for when n=1. As the expression gives 4 for this, we need to subtract 4 and add 1 (the value at the centre), giving 66917003.