Any odd number can be expressed as the difference of two perfect squares (idea)

I think that what you have noticed is another way of saying that for any positive integer n, n² is equal to the sum of the first n odd numbers, e.g. 3² is the sum of 1, 3, and 5, the first 3 odd numbers. While this fact seems odd when thought of this way, it makes perfect sense if you consider playing blocks with a kid.

Imagine, for a moment, that you have a huge pile of blocks and that you are making perfect squares out of them. You start with a 1x1 square.

          X

Now, to make it a 2x2 square you need to add 3 blocks

          XX
          XX

3x3 is just the same, but it now requires 5 additional blocks.

          XXX
          XXX
          XXX

In fact, to move from any nxn square to the (n+1)x(n+1) square requires the addition of 2n + 1 blocks. Coincidentally, this sequence of additional blocks happens to be all of the odd numbers.

I hadn't thought about this before from the direction you took, thanks for pointing it out to me.

Note: I am not a mathematician, nor do I play one on tv. I am a programmer entranced with the beauty of numbers and shapes. Like rabidcow, my mathematical experience is limited to the occasional breakthrough insight that I later learn was first noticed by Isaac Newton while he was still in diapers.