With the exception of the hyperpyramidal numbers (which include the pyramidal numbers and the centered pyramidal numbers, i.e. the (centered polygons) pyramidal numbers, as 3-dimensional hyperpyramidal numbers,) all the figurate numbers considered are regular polytope numbers corresponding to regular convex polytopes in a d-dimensional Euclidean space , d ≥ 0.
Among the hyperpyramidal numbers, the d-dimensional square hyperpyramidal numbers, although not corresponding to regular polytopes, are of particular interest since they are building blocks for the construction of the hyperoctahedral numbers (orthoplicial polytopic numbers), which are regular polytopes. For example, the nth octahedral number is the nth square dipyramidal number, i.e. it is the adjunction of the nth square pyramidal number to the (n-1)th square pyramidal number (corresponding to joined square pyramids at their square bases,) while for hyperoctahedral numbers of dimension d ≥ 3 we must do d-2 successive adjunction operations.
Otherwise, considering nonconvex regular (e.g. stellated) polytopic numbers or considering nonregular (e.g. Archimedean solids) polytopic numbers would open the door to a humongous number of possibilities...