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Well, after an embarrassing mistake, let me offer a heuristic I noticed that may be helpful. It seems when the above iff condition holds, the last $A$ value in $b(n)$ is equal to $8k^2+2$, and conversely when $A \not= 8k^2+2$ then $k \not\in A106483$.
Side note: Just because a Turing machine defines a state doesn't mean it is used. However, the definition of the problem says every tile (state) is used at least once, therefore, if there is a halting state, the machine will halt.
It might be possible to prove no such Wang tile set exists by constructing equivalent Turing machines. There's some technical difficulties and you have to make some particular assumptions (or at least be content with an answer in one quadrant of the plane). The key is the halting states; halting="does not tile plane". You start with at least one, and need to remove a state (tile) and end up with none (tiles the plane). So show after removing any single state (any single tile), you will have at least one more halting state than you want (except for the time you remove the state that will halt)
I would say there are a number of "accidental" discoveries that were made more obvious after visualization: mathoverflow.net/questions/178139/… . Perhaps some examples there should be included in this list.
