It appears that slight modification of the growth rules allows for synchronization of star-center turn-ons. This looks to be leading toward verification of a log-periodic Ulam Structure: at times $t = k^n, n \in 1, 2, \ldots$ the growth function goes extinct except for singular sources at each of five pentagonal corners (which alternate between even / odd $D_{10}$ axes as $n$ increases). The following animation goes through $n=1$ and $n=2$ with only a few sync errors:

Synchronization is important to long range order, since it can effectively prevent patterns of destructive interference from introducing wrong configurations. While mathematically perfect growth functions may not ever occur in the quasicrystal lab, they can certainly give us insight to what may happen there. The biggest takeaway (which is hinted at in Socolar's "Growth Rules for Quasicrystals") is that frustration is an important part of growing Ulam structures. Snowflakes tend to grow on corners due to sharpness of the electromagnetic field, so perhaps a similar mechanism affects quasicrystals.

Now that we have a perfectly synchronized function up to time $t \sim 100$, we can calculate the growth sequences fairly easily:

Length /@ AStates[[All, 2, 2]]
Differences[%]
Divide[%% - 1, 5]
Divide[%%, 5]
Transpose[Partition[%, 4]]

(* editor's note: these sequences have been corrected on Oct. 26 *)
Out[]:= {1, 1, 1, 1, 1, 6, 6, 6, 6, 16, 16, 16, 16, 21, 21, 21, 21, 36, 36, 
36, 36, 61, 61, 61, 61, 71, 71, 71, 71, 101, 101, 101, 101, 121, 121, 
121, 121, 136, 136, 136, 136, 156, 156, 156, 156, 176, 176, 176, 176, 
191, 191, 191, 191, 236, 236, 236, 236, 306, 306, 306, 306, 331, 331, 
331, 331, 396, 396, 396, 396, 441, 441, 441, 441, 471, 471, 471, 471, 
511, 511, 511, 511, 551, 551, 551, 551, 581, 581, 581, 581, 671, 671, 
671, 671, 736, 736, 736, 736, 751, 751, 756}
Out[]:= {0, 0, 0, 0, 5, 0, 0, 0, 10, 0, 0, 0, 5, 0, 0, 0, 15, 0, 0, 0, 25, 0, 
0, 0, 10, 0, 0, 0, 30, 0, 0, 0, 20, 0, 0, 0, 15, 0, 0, 0, 20, 0, 0, 
0, 20, 0, 0, 0, 15, 0, 0, 0, 45, 0, 0, 0, 70, 0, 0, 0, 25, 0, 0, 0, 
65, 0, 0, 0, 45, 0, 0, 0, 30, 0, 0, 0, 40, 0, 0, 0, 40, 0, 0, 0, 30, 
0, 0, 0, 90, 0, 0, 0, 65, 0, 0, 0, 15, 0, 5} (* note sync breaking *)
Out[]:= {0, 0, 0, 0, 0, 1, 1, 1, 1, 3, 3, 3, 3, 4, 4, 4, 4, 7, 7, 7, 7, 12, 
12, 12, 12, 14, 14, 14, 14, 20, 20, 20, 20, 24, 24, 24, 24, 27, 27, 
27, 27, 31, 31, 31, 31, 35, 35, 35, 35, 38, 38, 38, 38, 47, 47, 47, 
47, 61, 61, 61, 61, 66, 66, 66, 66, 79, 79, 79, 79, 88, 88, 88, 88, 
94, 94, 94, 94, 102, 102, 102, 102, 110, 110, 110, 110, 116, 116, 
116, 116, 134, 134, 134, 134, 147, 147, 147, 147, 150, 150, 151}
Out[]:={0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 5, 0, 0, 
0, 2, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 4, 0, 0, 0, 4, 0, 
0, 0, 3, 0, 0, 0, 9, 0, 0, 0, 14, 0, 0, 0, 5, 0, 0, 0, 13, 0, 0, 0, 
9, 0, 0, 0, 6, 0, 0, 0, 8, 0, 0, 0, 8, 0, 0, 0, 6, 0, 0, 0, 18, 0, 0, 
0, 13, 0, 0, 0, 3, 0, 1}
Out[]:= {
{0, 1, 2, 1, 3, 5, 2, 6, 4, 3, 4, 4, 3, 9, 14, 5, 13, 9, 6, 8, 8, 6, 18, 13}, 
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}

The first sequence is the number of suns turned on at time $t$. None of these integer sequences appear to be on OEIS, though many similar sequences are. Actually, the growth algorithm is fairly difficult to program, so it's more likely that OEIS would have something on the triality pattern itself, for example:

Length /@ NestList[Function[{verts}, Union@Join[Flatten[
      Nearest[Complement[AStates[[-1, 2, 2]] , verts ], #] & /@ verts,
       1], verts]], {{0, 0}}, 10]
Divide[% - 1, 5]
Differences[%]
Differences[%%%]

Out[276]= {1, 6, 16, 21, 36, 61, 81, 106, 136, 176, 211}
Out[277]= {0, 1, 3, 4, 7, 12, 16, 21, 27, 35, 42}
Out[278]= {1, 2, 1, 3, 5, 4, 5, 6, 8, 7}
Out[279]= {5, 10, 5, 15, 25, 20, 25, 30, 40, 35}

Neither do any of these appear in OEIS, so perhaps we are really doing something fun and original here.