[seqfan] Tiling binary numbers = A125121?

[Allan Wechsler]

The discovery of the "hat einstein" has me thinking about tiling again. I
apologize for the near-incoherence of the following explanation, but I hope
patient SeqFans will puzzle out my meaning.

What bit-patterns "tile"?

That is to say, what finite sets of integers can be used to "tile" the
integers? Each tile consists of a shifted copy of the prototile; in this
problem, reflections are not allowed.

For example, the bit string 100011 *does *tile the integers, with the
pattern:

...BAACBBDCCEDDFEE...

But 10011 does *not* tile, although if reflection were allowed (to produce
11001), it would tile.

You can identify finite bit patterns with binary numbers. I started
figuring out which patterns tile, and I decided it was simpler to allow
numbers that ended with 0. After enumeration a couple of dozen of these
"tiling binary numbers", the only remaining match in OEIS was A125121, the
"sturdy numbers". Are the sturdy numbers and the tiling binary numbers the
same? Is one a subsequence of the other?