## S000843

Half of all regular star-polytopes having n sides.

0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 2, 3, 0, 0, 1, 2, 3, 4, 0, 0, 2, 4, 1, 3, 0, 0, 1, 2, 3, 4, 5, 0, 0, 1, 2, 3, 4, 5, 6, 0, 0, 2, 4, 6, 1, 3, 5, 0, 0, 3, 6, 2, 5, 1, 4, 0, 0, 1, 2, 3, 4, 5, 6, 7, 0, 0, 3, 6, 1, 4, 7, 2, 5, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 0, 2

1

We say “half” because the other half consists of duplications of the first half, but in reverse order. Each polytope begins and ends with 0. For each n, we list all polytopes. For completeness, the first two entries are for a point (0,0) and a line (0,1,0). The book by Conway et al has a few drawings of these figures on page 404. All the polytopes are shown in the link below.

T. D. Noe, Plot of 939 terms (first 20 rows)

T. D. Noe, Table of 939 terms (first 20 rows)

T. D. Noe, Plots of the 65 star-polytopes

John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss, The Symmetries of Things, A. K. Peters, 2008.

(Mma) Flatten[Join[{0, 0}, Table[d = Flatten[Position[Table[GCD[i, n], {i, n/2}], 1]]; Table[Table[Mod[j*i, n], {i, 0, n}], {j, d}], {n, 2, 10}]]]

Cf. A023022, A182972.

nonn,tabf

T. D. Noe, Feb 02 2016