[seqfan] Re: Tiling binary numbers = A125121?

[M. F. Hasler]

It's not completely clear to me what you mean by tiling. Do you mean that
you can get "all 1's" in binary from copies of the number n
shifted to the left such that the copies do not have overlapping 1's ?
As for 5 = 101,
101 + 1010 + 101 0000 + 1010 0000 + ... ?

On Thu, Apr 13, 2023, 22:35 Allan Wechsler <acwacw at gmail.com> wrote:

> 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?
>
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