I don't suppose it has anything to do with binary? I can see a pattern in the 2^1 and 2^2 columns (they're always 0). I don't see anything to do with symmetry though.
When I mentioned that we might have to do something with the logarithm of c in binary form, it was because you mentioned that we're meant to be looking for binary patterns. If the only reason it isn't O(1) is because of square roots, where should we be looking in binary?
Oh shiiiiiit I just noticed that it's following the same pattern as the triangle numbers from here >>5263 with the length of the 0s being 2, 3, 2, 4, 2, 3, 2, 5, 2, 3, 2. It's the same pattern but with a different difference each time. Here it's always a difference of 1. In the triangle numbers, it's always the next triangle number (so ^#2, ^#3, ^#2, ^#4 so on). "Coincidences", am I right? If that's not what you wanted us to find, that's still pretty crazy.