Hi, I just saw the table you added. Thanks, this is really cool!

If I may ask, what was your procedure for finding a decomposition?

Whoa! I guess shifting every square left and down as soon as they are generated cannot be done without loss of generality. Thanks for this!

@DustinG.Mixon Thanks :) I split $x_i$ into $\sqrt {x_i}\cdot\sqrt {x_i}$ like a doofus. By the way, simple as your comment is, if you add it as an answer, I will happily accept it, as this is what I was looking for!

@DustinG.Mixon Thanks, the lower bound argument makes sense. Could you spell out the upper bound, sorry? Perhaps I'm botching the C-S inequality, but I can only get $\sum_{i=1}^n x_i\leq n^2$.

@SamHopkins They are supposed to be the same $n$. The peculiar case is $n=8$, when only $7$ squares are needed to produce a maximal sum. Of course, we can artificially decompose one of the squares into two smaller ones, giving us $8$ squares, as prescribed.

Trivial update (posted here to avoid spamming the homepage, hopefully): $f(11) = 35$.

@GerryMyerson Yes. Sorry if I'm not being very formal with the parameters of the question, but indeed the squares have to be sort of "snapped to" the integer coordinates of the plane.