[seqfan] Re: Integral sets of points (A007285, A096872 and A096873)
Sascha Kurz
sascha.kurz at uni-bayreuth.de
Thu Nov 17 07:26:59 CET 2011
I agree with Robert. To keep it consistent it would not add a(1), a(2) to
A096872 and A096873 either.
It would be indeed more precise to state that not all points are on a line
and to explain the term "integral set".
Even the existence of A096873(8) is an open problem.
If the suggested three additional sequences are considered to be
interesting I may compute or look up the values and submit them.
BTW: Some literature may be added, e.g.
A007285,A096872:
* S. Kurz and A. Wassermann: On the minimum diameter of plane integral
point sets, Ars Combinatoria, Vol. 101 (2011), Pages 265-287
A096873:
* T. Kreisel and S. Kurz: There are integral heptagons, no three points on
a line, no four on a circle, Discrete and Computational Geometry Vol. 39
Issue 4 (2008), Pages 786-790
* J. Solymosi and F. de Zeeuw: On a Question of Erdős and Ulam,
Discrete and Computational Geometry Vol. 43 Issue 2 (2010), Pages 393-401
Best wishes,
Sascha
Am Do, 17.11.2011, 06:21, schrieb israel at math.ubc.ca:
> For A007285, you can't have 1 or 2 points "not all on a line", so I don't
> think there should be a(1) or a(2)..
>
> Robert Israel israel at math.ubc.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia Vancouver, BC, Canada
>
>
> On Nov 16 2011, Benoît Jubin wrote:
>
>
>> Dear Seqfans,
>>
>>
>> I think a few initial values should be added/corrected in the
>> sequences A007285, A096872 and A096873, and that the definitions could be
>> modified to treat them in a uniform way. I propose the following; can
>> you confirm this is correct?
>>
>> A007285: Minimum diameter of an integral set of n points in the plane,
>> not all on a line. add: a(1)=0 and a(2)=1 (therefore modify offset to 1)
>>
>>
>> A096872: Minimum diameter of an integral set of n points in the plane,
>> no 3 on a line. modify: a(1)=0
>>
>>
>> A096873: Minimum diameter of an integral set of n points in the plane,
>> no 3 on a line, no 4 on a circle. modify: a(1)=0
>> add comment: As of 2011, it is not known if this sequence is finite.
>>
>> For all three sequences: add the definition: "An integral set is a set
>> where all distances between points are integers." add the link:
>> http://ginger.indstate.edu/ge/COMBIN/GEOMETRY/intdistance.html
>>
>>
>> Add three analog sequences where all points are furthermore required
>> to have integral coordinates?
>>
>> Thanks,
>> Benoit
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