Golomb ruler sequences

[David Wilson]

For those interested in Golomb rulers,

Part of the problem is that Golomb rulers are expressed in at least three
different ways in the OEIS and MathWorld.  This may be consistent with
the sources, but it is confusing to the reader.  The three ways I have seen
are:

(1) Zero-mark representation: List the positions of the marks on the ruler,
the first mark being 0.  This is what is done in the MathWorld article
"Golomb ruler".
(2) One-mark representation: List the positions of the marks on the ruler,
the first mark being 1.  This is what is done in A031873 and similar
sequences.
(3) Distance representation: List the distances between adjacent marks.
This is what is done in A039953.

I personally tend to like zero-mark rep, because:

- Zero-mark is consistent with the MathWorld definition.
- With zero-mark and one-mark, the number of marks is simply the
number of elements, with distance rep, it is the number of elements + 1.
- With zero-mark, the length of the ruler is simply the last element.  With
one-mark, it is the last element - 1, with distance rep, it is the sum
of all elements in the sequence.
- With zero-mark and one-mark, distances between two marks is simply
the absolute difference of the two marks.  With distance rep, it is the
sum of several adjacent elements that lie between the marks.

At any rate, there are several Golomb ruler sequences which probably
need to be revisited.  These are

A031869 through A031875.  These were originally all rubber stamps of
A031863, which NJAS just modified for clarification.  They give Golomb
rulers of 6 through 12 marks, in one-mark representation.  I suggest
that they be converted to zero-mark representation, and new sequences
added so that we have Golomb rulers with marks from 4 through the
largest known value, which I think is currently 22.

There is also A039953 which gives a triangle of Golomb rulers in
distance rep.  Since this triangle apparently comes from a reference,
we should leave it in distance rep and make it clear that this is the
representation.  We might create another triangle in zero-mark rep.
(By this I mean I will do the work if NJAS agrees to my assessment).

There are a few other scattered sequences related to Golomb rulers
that might be revisited as well.