JCSBimp's Hexagonal Orthogonal

I am ironing out some new calculations that may come to nothing, but also may show me an advance into simplification.

I was looking at the fact that a series of consecutive values of r_n in any odd positive integer m’s Remainder Series will differ from a series of consecutive perfect squares by linearly-progressing amounts.

I am charting this behavior both for some of the discrete semiprimes I know very well and for some randomly-generated ones, in order to see the behaviors in which I’m interested from both the known answer case and the unknown answer case.

And, as always, I’m happy to share my digital scribble sheet: