[seqfan] Re: A possible characterization of A125121

[Allan Wechsler]

I think Don Reble has a counterexample, so my conjecture is false, and I
will be preparing a new sequence for submission.

David Wilson's proposed counterexample, 35, is 100011 in binary, and tiles
like this:

...BAACBBDCCEDDFEEG...

In other words, the "quotient" is ...100100100100...

Although some sturdy integers don't tile, I am left with the weaker
conjecture that all tiling integers are sturdy, and would welcome a
counterexample to that too.


On Sat, Nov 16, 2013 at 12:35 AM, David Wilson <davidwwilson at comcast.net>wrote:

> Explain how 35 tiles.
>
> > -----Original Message-----
> > From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Allan
> > Wechsler
> > Sent: Friday, November 15, 2013 1:27 PM
> > To: Sequence Fanatics Discussion list
> > Subject: [seqfan] A possible characterization of A125121
> >
> > Are these also exactly those integers whose binary representations "tile"
> > the integers?
> >
> > Represent an integer N as a set of indices S={i} such that sum 2^i = N.
> >  For any N we can ask: does there exist a family of shifted copies of S
> which
> > are (a) disjoint, and (b) whose union covers the integers?  Each N poses
> a
> > pleasant little tiling problem.
> >
> > I amused myself by listing the integers that tile, and found that it
> matched
> > A125121, whose given characterization also involves binary
> representations,
> > but seems quite different from my tiling description.
> >
> > Can anyone prove that A125121 contains exactly the integers that tile, or
> find
> > a counterexample?
> >
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