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Classifications of figurate numbers
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 -dimensional Euclidean space
Among the hyperpyramidal numbers, the -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 th octahedral number is the th square dipyramidal number, i.e. it is the adjunction of the th square pyramidal number to the th square pyramidal number (corresponding to joined square pyramids at their square bases), while for hyperoctahedral numbers of dimension we must do 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...