[seqfan] Re: Tiling binary numbers = A125121?

[David Radcliffe]

A subset S of Z is said to tile the integers if and only if Z is the union
of disjoint translates of S, i.e. there exists a set T such that every
integer can be expressed uniquely as s + t for some s in S and t in T. The
problem of characterizing the finite subsets that tile the integers appears
to be unsolved, although some partial results are known.

Here's one reference:
Coven, Ethan M., and Aaron Meyerowitz. "Tiling the integers with translates
of one finite set." *Journal of Algebra* 212.1 (1999): 161-174.
https://arxiv.org/abs/math/9802122



On Thu, Apr 13, 2023 at 9:34 PM 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?
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>