A picture of the Sparse ruler pattern, A046693(n) - A309407(n) to n=4443, is at the bottom of  https://community.wolfram.com/groups/-/m/t/1749574     It's a fascinating pattern. "Dark Satanic Mills on a cloudy day" -- NJA Sloane The green squares are likely all black with optimal sparse rulers.   Many of the black squares are likely white.   The pattern is getting so weird that I'm not sure what to predict now.   --Ed Pegg Jr