Convex-octagon and concave-quadrilateral coincidence

Determine whether the equality between the maximum region counts for n convex octagons and n concave quadrilaterals has a geometric explanation or is merely a coincidence.

Background

The paper observes that the sequence for the maximum number of regions formed by n convex octagons agrees with the sequence previously associated with n concave quadrilaterals. The two constructions appear geometrically unrelated.

The authors explicitly question whether a structural explanation connects the two problems or whether the identical formulas arise accidentally.

References

Is it just a concidence tht the maximum number of pieces that can be obtained with $n$ (convex) octagonal cuts is the same as can be obtained with $n$ concave quadrilateral cuts? We do not know!

— Cutting a Pancake with an Exotic Knife  (2511.15864 - Cutler et al., 19 Nov 2025) in Section 9.2, caption to Figure 7 and Section 10, item 5