The sequence of polygonal numbers isn't (at least as I see it) as simple as a number n for f(x) where f(x)=f(-x). It relies upon the union of f(x-1) and the (unfilled) polygon g(x) that is the perimeter extant of f(x), and the special case of f(0) being 1, so that for squares length 1 sides is 4x1 minus L0's 1 (added to L1's 1), length 2 is 4x2 minus the three coincident L2 spots (added to L2's total of four), length 3 is 12-5(+9), etc.

(Noting that I'm using length of side

x that is actually one less than the real number used, which is "fenceposts, not fence-bars", just to save fuss in this segment.)

The transition up the negatives (towards zero) is therefore different in principle from backtracking down the positives (towards zero) and just reversing the sign of both. Indeed (even after readjusting for my adjust-by-1 'error') it can be seen that a positive output number comes out of the negative square. The simpler, in this case, y=x² analogue to the more complicated sum of 1,3,5,7,9,… makes it obvious (much as adding -n,-n+2,-n+4 works on the flip-side, and …-1,+1,… nicely bridges across the central pillar of zero), but for the less trivial y=NPolygonOf(N,x) for N≠4 it'd be a funny Union¹ of old full-polygon and new edge-polygon to make the set deplete.

What I'd find interesting is the efficacy of negative

sides (or zero, if not 1 or 2 also), then fractional and eventually complex+ numbers in either/both the N and x positions (for N=4, 'otherworldly' values of x can be compared against the existing known treatment of 'just squaring them', as a first test for reality). Assuming we aren't going to disallow them (as an incalculable singularity/undefined value) or lazily neuter them into just counting numbers (as per the negatives in the lazy way of handling them) into accepted order.

But minds more learned than mine² have no doubt applied themselves to these issues, even if I'd spent more than just the half an hour on my own analysis/reinventing-the-wheel!

¹ Probably just means that one should

not think of it in terms of sets, but I'm not entirely convinced we should just use |x| in the equation and disavow the handling of negative numbers in the series by glossing over them.

² I

think I would be no false modesty to count Euler amongst that number.