The old n^2+1 uses a n*(n+2) grid, but the n+2 get's hidden when you look at the symmetric version. I hope that calculating a non-symmetric version of the new weight may show an structure that is easier to understand and generalize.
Also increasing the grid may be helpful to understand which are atoms, if some ideal atoms get split to fix in the grid, and if the bars in the center are real or an artifact. (Is there a 45° rotated square in the center?). So this also may help to understand and generalize the result. But the run time grows very fast for big grids, so I'm less optimistic about this possible improvement.
Some of these point arrangements superficially resemble Chladni figures (https://chalkdustmagazine.com/regulars/on-the-cover/cover-ch...) coming from the vibrations of a square drum.
Orthogonally, I wonder if these arrangements are generalizable to give lower bounds for N = n^2 + 1
The old n^2+1 uses a n*(n+2) grid, but the n+2 get's hidden when you look at the symmetric version. I hope that calculating a non-symmetric version of the new weight may show an structure that is easier to understand and generalize.
Also increasing the grid may be helpful to understand which are atoms, if some ideal atoms get split to fix in the grid, and if the bars in the center are real or an artifact. (Is there a 45° rotated square in the center?). So this also may help to understand and generalize the result. But the run time grows very fast for big grids, so I'm less optimistic about this possible improvement.
None of the images has any squares.
The n=17 case is a famous math meme.
But where is the picture for this lower bound packing?