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

Allan Wechsler acwacw at gmail.com
Mon Nov 24 04:08:24 CET 2014


A219529 is the coordination sequence for the 3.3.4.3.4 network.

On Sun, Nov 23, 2014 at 3:31 PM, Neil Sloane <njasloane at gmail.com> wrote:

> Brad, I was just about to reply when Maurizio's message arrived. All the
> terms that you and he found can be explained
> by the (conjectured) generating function
> (x^2+x+1)*(x^4+3*x^3+3*x+1)/((x^4+x^3+x^2+x+1)*(x-1)^2),
> which is nice and symmetric.
>
> Finding a proof should be easy, now we know the answer.
>
> I'll update the entry (A250120) later today.
> And I'll certainly include your picture, which
> is far nicer than mine. (I assume that's OK?)
> I'll include both pictures, yours and mine.
>
> 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
> >>
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> >>
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