[seqfan] Re: Tiling binary numbers = A125121?

[hv at crypt.org]

I believe we're talking about tiling the integers from -inf to +inf.
The original "100011" example does not tile Z+, but does tile Z.

The "tile" is a template which "covers" the places where there is
a 1 in the binary expansion, and does not cover where there is a 0.
Placed at any integer position n, the "100011" tile covers n, n+4, n+5.

Hugo

"M. F. Hasler" <oeis at hasler.fr> wrote:
: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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:>
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