[seqfan] Re: Tiling binary numbers = A125121?

[Allan Wechsler]

Here is a rigorous definition of a binary tiling number.

Let S be a *finite* set of integers. If k is an integer, then when I say
S + k, I mean a new set of integers, whose elements are derived from the
elements of S by adding k to each. If S + k = S', I say that S and S' are
congruent.

If Z can be covered by a *disjoint* family of sets congruent to S, then I
say that S tiles Z.

So far, I have only said what it means for a set to tile Z. What does it
mean for a number to tile? Express n as a sum of distinct powers of 2,
2^e0 + 2^e1 + 2^e2 + ... + 2^e[k], with e[i] = e[j] only if i = j. Then I
say that n is a binary tiling number if S = {e0, e1, e2, ... e[k]} tiles Z.

This differs from Maximilian's guess, because he requires that S exactly
tile only the *nonnegative* integers. Maximilian's condition is stricter
than mine, because one can prove (using a state-machine argument) that if S
tiles the nonnegative integers, it must tile Z, whereas the counterexample
of 100011 (decimal 35) shows that sets that tile Z do not necessarily tile
N.

I just checked to see if the *odd* binary tiling numbers were in OEIS.
They're not.

On Sat, Apr 15, 2023 at 9:50 AM Fred Lunnon <fred.lunnon at gmail.com> wrote:

>   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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>
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