[seqfan] Re: [The Tiling List] Re: Coordination sequences for planar nets

Mon Nov 24 07:38:23 CET 2014

Neil, thanks for citing me in A250120.

However, what I don't get is why you resort to the second difference,
since already the *first* difference show the periodicity 5:
4 (4 6 4 5 5) repeated.

In any case there is a mistake in your "FORMULA" section, since the
period of the second difference is (2 -2 1 0 -1), its length is of course
5, not 4.

Best regards,
Maurizio

On Mon, Nov 24, 2014 at 12:36:33AM -0500, Neil Sloane wrote:
> I've fully updated A250120 - thanks to everyone for their contributions.
> 
> The full list of sequences for the 11 planar nets now looks like this:
> 
> List of coordination sequences for uniform planar nets: A008458 (the planar
> net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and also apparently
> 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706 (3.3.3.4.4), A072154
> (4.6.12), A219529 (3.3.4.3.4), A250120 (3.3.3.3.6), A250122 (3.12.12).
> 
> Darrah, I just created the last one in your name (mostly to stake your
> claim to it, also to complete the list). Of course, add more material there
> as it develops.
> Updates to any of these entries are welcomed.
> 
> Best regards
> Neil
> 
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
> 
> 
> On Sun, Nov 23, 2014 at 3:04 PM, Brad Klee <bradklee at gmail.com> wrote:
> 
> > Hi Neil,
> >
> > I checked A250120 by looking at your drawing. That's a cool drawing, and
> > it reminds me of the Gosper Island tiling.
> >
> > Just by looking at your picture, it seems like your first few numbers are
> > correct.
> >
> > I also wrote a computer program to extend the sequence. The algorithm
> > recursively enumerates points in counted subset by expanding around each
> > currently included point using the six hexagonal generators. Then a filter
> > removes any duplicate vertices and vertices belonging to the uncounted
> > lattice with Sqrt[7] spacing.
> >
> > This accidentally introduced another unrecorded sequence
> >
> > 1, 6, 15, 30, 49, 73, 102, 135, 174, 217, 265 ...
> >
> > Which is just the total number of points covered. The sequence given by
> > the first derivative is your counting sequence
> >
> > 1, 5, 9, 15, 19, 24, 29, 33, 39, 43, 48 ...
> >
> > The sequence given by the second derivative is another unrecorded sequence
> >
> > 4, 4, 6, 4, 5, 5, 4, 6, 4...
> >
> > The first sequence approximately gives the area, the second approximately
> > gives the perimeter, and the third seems to be bounded above by 2 Pi.
> > Compare this to sequence of derivatives of circular area
> >
> > Pi R^2, 2 Pi R, 2 Pi, 0 , 0 ...
> >
> > In this case there is something weird happens along the boundary, so there
> > is a sequence for third derivative
> >
> >  0, 2, -2, 1, 0, -1, 2, -2 ...
> >
> > But it appears that this sequence will have a zero average in the limit
> > where the number of terms N approaches infinity. Maybe this palindrome
> > pattern continues? Up to 20 terms, your sequence A250120 appears to be the
> > third integral of a periodic pattern.
> >
> > The code I give is probably not the best way to specify this sequence. It
> > should be easier to find a recursion for the second or third derivatives
> > because those sequences seem to have a finite alphabet.
> >
> > Thanks,
> >
> > Brad
> >
> >
> >
> > On Sun, Nov 23, 2014 at 12:59 PM, Neil Sloane <njasloane at gmail.com> wrote:
> >
> >> There are 11 uniform (or Archimedean) tilings in the plane.
> >> If we take the 3^6 tiling (or net) (6 triangles around each point),
> >> start at a lattice point, and walk outwards for 0, 1, 2, 3, 4, ... steps,
> >> the number of points we reach for the first time gives the sequence 1, 6,
> >> 12, 18, 24, 30, 36, 42, ...,
> >> increasing by 6 at each step after the first.
> >> This is sequence https://oeis.org/A008458 in the OEIS.
> >> (In other words, it is the number of nodes at graph distance n from a
> >> fixed
> >> node.)
> >>
> >> The planar net 3.6.3.6 gives https://oeis.org/A008579, and I just added a
> >> primitive drawing to the entry to illustrate the first few terms. This is
> >> rather more complicated.
> >>
> >> Next I looked at the 3^4.6 net, and for the initial terms of
> >> the sequence I get 1,5,9,15,19,24, by hand.
> >> This is bothersome, because (a) it is quite irregular, and (b) it was not
> >> in the OEIS!  I just added it (https://oeis.org/A250120), along with a
> >> drawing showing my calculations. I have no confidence in these numbers -
> >> could someone check them?
> >>
> >> I don't know how many of the other planar nets are in the OEIS. 3^6 is
> >> A008458, 3^4.6 is tentatively A250120, 3^3.4^2 is A008706, 3^2.4.3.4 = ?,
> >> 4^4 is A008574, 3.4.6.4 is ?, 3.6.3.6 is A008579, 4.8^2 is A008576, 6^3 is
> >> A008486, and the others I don't know.
> >>
> >> Best regards
> >> Neil
> >>
> >> _______________________________________________
> >>
> >> Seqfan Mailing list - http://list.seqfan.eu/
> >>
> >
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