JCSBimp's Hexagonal Orthogonal

Let m be an odd positive integer, c its Adjusted Ceiling Root, and r the difference between c squared and m.

By the Adjusted Ceiling Root of m, I mean the least positive integer whose odd/even parity is the same as (m+1)/2 whose square is greater than m. I call that perfect square the Adjusted Ceiling Square of m.

I just noticed what I think some elementary calculation would bear out: that the value of r is always congruent to 0 or 1 modulo 4.

This may help guide my thoughts on figuring out a connection between the value of m and its “Series of Ceiling Squares” as it has encountered some trickiness when I try to deal with even remainders.

In any case, it’s something I only newly noticed.