[seqfan] Re: A possible characterization of A125121

[Allan Wechsler]

I think I am having a terminology difference with David Wilson. I intended
my "tilings" to cover all the integers, not just the non-negative ones, and
thus right shifts are not only allowed, they are necessary.

This also explains our difference of opinion about 35.  I agree 100011
can't tile just the non-negative integers, but it *can* tile all the
integers, as I thought I showed in a previous message.


On Sat, Nov 16, 2013 at 5:16 PM, David Wilson <davidwwilson at comcast.net>wrote:

> If you allow only left shift, the tileable numbers are A064896.
>
> If you allow left and right shift, the tileable numbers are the numbers of
> the form 2^m * A064896(n).
> This sequence is a strict subset of the sturdy numbers A125121, with 35 the
> first non-tileable sturdy number.
>
> If you allow left shift and reversal, the tileable numbers are all odd, and
> the first non-tileable odd number is 27.
>
> If you allow left shift and right shift and reversal, the tileable numbers
> are 2^m * elements of the previous sequence.
>
> None of the last three sequences is in the OEIS.
>
> > -----Original Message-----
> > From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of David
> > Wilson
> > Sent: Saturday, November 16, 2013 12:36 AM
> > To: 'Sequence Fanatics Discussion list'
> > Subject: [seqfan] Re: A possible characterization of A125121
> >
> > 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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