[seqfan] Re: Tiling binary numbers = A125121?

[Fred Lunnon]

  Glad somebody else was sufficiently brave to ask!    WFL


On Sat, Apr 15, 2023 at 2:21 PM 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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