A Puzzle for Christmas Time

In my recent post about the sum of squares of the natural integers, I mentioned that one of the  problems of recreational mathematics  related to the series was the problem of counting the number of squares that were in an nxn square array such as the one above. 

Joshua Zucker sent an interesting (read more challenging) version of the problem of counting squares by suggesting that instead of an array of squares, we use an array of dots, like the one below.

This kind of array can be more challenging because in addition to squares like the ones below





You can get some turned like this:


And when the array gets a little larger, they start to have some like this:
 

Since Joshua has had time to think about this and come up with one of his (always) clever answers that makes it look easy (easier?).  I will leave this posed as a problem, and invite him to send his approach which I will post at a later date so as not to spoil the fun for the rest of us.  I already see that this will go off into some special partitions of integers which means I have to really put my thinking cap on.

In the meantime, enjoy the puzzle and send your responses for solutions for given sizes, or a general method.  I'm off to find paper and pencil for a snowy afternoon of mathematical doodling.

A puzzle for you all, from Santa Joshua.  Merry Christmas.

Joshua also asked another question in his comment.  Is there a geometric visualization for WHY the 2N+1 shows up in the formula.  I would love to hear and share your answers about that one as well.