[seqfan] Re: Tiling binary numbers = A125121?

[M. F. Hasler]

Yes of course, sorry for not mentioning the negative powers,
my main focus was to be sure about the basic concept.
The formulation in terms of disjoint union of translated sets seems the
most natural and elegant,
and the preprint David pointed to probably contains lots of nontrivial
results.

Then the "tiling binary numbers" are the sums of powers of 2 of the
elements of such sets,
and the odd integers of that form, which Alan considered, correspond to the
"canonical representative" of these sets, translated such that min S = 0.

I reckon that this should start, in binary:
1, 11, 101, 111, 1001, 1111, 10001, 10101, ...
In particular, all integers of the form 2^n +- 1 are in that set, and also
all those of the form 1(01){n} and 11(0011){n}, etc.
i.e., 1{k}(0{k}1{k}){n} in regex syntax.
We might call nontrivial the tiling numbers which are not of such a simple
form.
Probably I have overseen a slightly more general simple periodic pattern
that could be included among the "trivial" solutions...
Oh yes, we might consider "trivial" all those for which a finite disjoint
union of translates yields a number of the form 2^n-1, corresponding to a
set S={0...n-1}.
100011 would then be a nontrivial solution. But it can be obtained by
"rotating" 111000 two places to the left. Obviously the first rotation,
110001, would also work.
 Maybe all these should be considered trivial, too?

-M.

On Sat, Apr 15, 2023, 10:24 <hv at crypt.org> wrote:

> 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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