In my work on lattice point enumeration of polytopes, I stumbled upon the following sequence: \begin{eqnarray} 1, 120, 579, 1600, 3405, 6216, 10255, 15744, 22905, 31960, 43131, ... \end{eqnarray} which counts the Structured great rhombicosidodecahedral numbers (A100145) by the formula \begin{eqnarray} a(n)=\tfrac{1}{6} (222 n^3-312 n^2+96 n). \end{eqnarray} Such numbers fall into the category of figurate numbers, which count the number of points in a sequence of similar discrete geometric shapes. For example, the triangular and square numbers bear their names because they count the dots arranged in a sequence of triangular $(1,3,6,10,...)$ and square $(1,4,9,16,...)$ configurations. One generalizes these to higher dimensional regular polyhedral numbers like tetrahedral (A000292) or dodecahedral (A006566) numbers, for instance. These numbers are always enumerated by $\mathbb{Q}$-polynomials of degree $n$, where $n$ is the dimension of the polyhedron.
For the sequence above, the author gives the following description:
Structured polyhedral numbers are a type of figurate polyhedral numbers. Structurate polyhedra differ from regular figurate polyhedra by having appropriate figurate polygonal faces at any iteration, i.e. a regular truncated octahedron, n=2, would have 7 points on its hexagonal faces, whereas a structured truncated octahedron, n=2, would have 6 points - just as a hexagon, n=2, would have. Like regular figurate polygons, structured polyhedra seem to originate at a vertex and since many polyhedra have different vertices (a pentagonal diamond has 2 "polar" vertices with 5 adjacent vertices and 5 "equatorial" vertices with 4 adjacent vertices), these polyhedra have multiple structured number sequences, dependent on the "vertex structures" which are each equal to the one vertex itself plus its adjacent vertices. For polystructurate polyhedra the notation, structured polyhedra (vertex structure x) is used to differentiate between alternate vertices, where VS stands for vertex structure.
At first read, this doesn't make any sense. I thought the regular truncated octahedron had 6 vertices at each hexagonal face, not 7 as the author claims. (I know that this sequence isn't bogus because I can generate it in a completely different context, that of computing the cohomology and geometric genera in a singularity theory problem.)
Can anyone make sense of this and help me understand the difference between regular and structured polyhedra?
Update (4-1-11): I emailed the author of the entry on OEIS and never heard back from him. I think the responsibility now lies with us to figure this out.