The triangular table view is fascinating. It looks like the periodic table. I wonder if there are number-theoretic lemmas (or at least conjectures?) about what "family" the optimal packing for a given number falls into (like diamond, diagonal strip, two blobs, etc). I didn't see anything when skimming the survey paper linked at the bottom of the site, but I'm sure there's a lot more literature here.

Looks like Hiroshi Nagamochi did all the boring work.

In case you want a challenge, 11 is the smaller that has a solution that has not been proven to be optimal.