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There is, I believe, a sequence of polyhedra whose shape approaches that of the icosahedron (they all have twelve pentagonal faces and the rest hexagons), and starts:

regular dodecahedron (C$_{12}$, dodecahedrane)

truncated icosahedron (C$_{60}$, buckminsterfullerene)

?: (C$_{240}$, buckminsterfullerene)

?: (C$_{540}$, ?)

I am interested in the polyhedra that these molecules are associated with, namely:

  • What is the correct name for this sequence of polyhedra? (oeis.org doesn't have any sequences that match "12,60,240,540" at present)
  • Is there a formula in the literature for the volume of these polyhedra? (the surface area is not a difficult computation)
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