Let's try to count the number of tapertopes, not including any round shapes. A formula for this list could lead to a formula for the tapertopes in general.
1D:
line = 1
2D:
square = 11
triangle = 1'
3D:
tetrahedron = 1''
square pyramid = 11' (Rob's notation) or 1'1' (my notation)
triangular prism = 1'1
111 = cube
4D:
1''' = pentachoron
11'' or 1''1'' = square dipyramid = 11 -> 0 -> 0
111' or 1'1'1' = cube pyramid = 111 -> 0
1'11 = triangular diprism = 1' x 1 x 1= 111 -> 11
1'1' = triangular prism pyramid = (1' x 1) -> 0 = 1'' -> 1' = (11->0) -> 1
1''1 = tetrahedral prism = 1'' x 1 = (1' x 1)
1111 = tetracube
There might be more 4D shapes. We should probably check that none of the 4D shapes are curved like the wedge. So far, our sequence is 1, 2, 4, 7.