Adam,
It looks like you really enjoy Number Theory. What you are talking about is well documented and people have been working with squares since the beginning of civilization (remember the Pithagorean theorem). Get a book about the history of Math or Number theory, you will enjoy it.
The pattern that you are talking about is called "The Difference of Squares". If you have two whole numbers X and Y then
(X^2) - (Y^2) = (X-Y)(X+Y)
If you have two consecutive numbers then (X-Y)=1, so the difference would be the sum of the numbers. Hope that this helps.

Robert