From Martin Gardner's 'From Penrose Tiles to Trapdoor Ciphers'

From page 14, Chapter 1; https://www.maa.org/sites/default/files/pdf/pubs/focus/Gardner_PenroseTilings1-1977.pdf

"Any spoke of the infinite cartwheel pattern can be turned side to side (or, what amounts to the same thing, each of its bow ties can be rotated end for end), and the spoke will still fit all surrounding tiles except for those inside the central cartwheel. There are 10 spokes; thus there are 2^10 = 1024 combinations of states. After eliminating rotations and re- flections, however, there are only 62 distinct combinations. Each combi- nation leaves inside the cartwheel a region that Conway has named a "decapod." Decapods are shapes of enlarged half darts. The decapods with maximum symmetry are the buzzsaw and the starfish shown in Figure 12. Like a worm, each triangle can be turned. As before, ignoring rotations and reflections, we get 62 decapods. Imagine the convex vertexes on the perimeter of each decapod to be labeled T and the concave vertexes labeled H. To continue tiling, these H's and T's must be matched to the heads and tails of the tiles in the usual manner. When the spokes are arranged the way they are in the infinite cart- wheel pattern shown, a decapod called Batman is formed at the center. Batman (shown in dark gray) is the only decapod that can legally be tiled. (No finite region can have more than one legal tiling.) Batman does not, however, force the infinite cartwheel pattern. It merely allows it. Indeed, no finite portion of a legal tiling can force an entire pattern, because the finite portion is contained in every tiling.'

The problem I am having is getting the 62. I thought that putting 10 triangles meeting at a vertex with an angle of 2pi/10 where the triangles can be oriented in 2 ways would be the same as a 2 colour bracelet with 10 beads.

Here are some OEIS sequences;

A006245 rhombus tiling of 2n-gon A(3)=2 A(4)=8 A(5)=62 A(6)=908 A(7)=24698

A000031 2 colour bracelets A(10)=46

A000029 2 color necklaces A(10)=60

Now A006245 gives the correct number for the term,but this is the number of tilings of a 2n-gon , not the number of Decapods.

I would have thought A000031 was the correct object but a(10)= 46, not 62.

So I must be misunderstanding, what I am doing wrong?

Stuart