[seqfan] Sparse Rulers and Golomb Rulers in two dimensions

Rainer Rosenthal r.rosenthal at web.de
Fri Aug 14 23:14:59 CEST 2009

The term "sparse ruler" is used in a comment from
Ed Pegg in A003022:
Comment from Ed Pegg, Jr. (ed(AT)mathpuzzle.com), Aug 17 2007:
If you're looking for something practical, which can measure
any distance, you need a "sparse ruler". David Fowler has
studied these (see link below).
But ... there is no such "link below". Has it been replaced
or deleted?

I would like to add the 2-dimensional versions of sequences
related to Golomb Rulers (A003022) and Sparse Rulers (A004137).
I think it would be a good idea to mention the term "sparse
ruler" in A004137 and the related sequences.

For the 2D-version, there is sequence A047800, counting all
the distances in an nxn grid. Define a nxn "Golomb Planeal"
as largest subset of the nxn grid where all distances are
different. Define a nxn "Sparse Planeal" as smallest subset
of the nxn grid with all its distances (i.e. A047800(n-1) many).
Computing the sizes will yield nice sequences, wouldn't it?

I did not find such sequences, but may be I haven't been looking
carefully enough. The idea for this comes from the ongoing
Al Zimmermann contest http://www.azspcs.net/Contest/PointPacking

(1) Fowler's "link below" in A003022 is missing.
(2) The term "sparse ruler" should be mentioned in
    seqence A004137 and related ones.
(3) Based on A047800 there are 2D variants possible
    for Golomb-like and for sparse rulers. I would like
    to know if there are already sequences dealing with
    that subject.


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