[seqfan] Re: A347521 Number of polyominoes with n cells formed by coordinates that are not coprime

[Allan Wechsler]

Thanks for the coordinates of the X-pentomino ("2,2-dimethylpropane",
[272]). To clarify, (104,5) isn't in the pentomino, but it's at the low
corner of its bounding rectangle. The center of the cross is at (105,6).
(It makes sense to find novelty at coordinates that have lots of prime
divisors but only one in common.)

On Thu, Sep 15, 2022 at 8:46 AM John Mason <masonmilan33 at gmail.com> wrote:

> Hi
> I am now in touch with the author.
> In the meantime I was able to understand the sequence - I admit I was a bit
> slow.
> The third pentomino is a cross, just outside the diagram on page 1 of the
> link (coordinates 104,5).
> Thanks to everyone for their help
> john
>
> On Thu, Sep 15, 2022 at 4:42 AM Allan Wechsler <acwacw at gmail.com> wrote:
>
> > I'm pretty sure the author means "maximal", in the sense that for a
> > polyomino to be included, it has to occur completely bordered with white
> > cells.
> >
> > This explains the first few entries. For example, the monomino clearly
> > occurs, and the domino clearly doesn't. Proving that there are no
> isolated
> > dominoes is slightly tedious but I'm perfectly willing to believe that
> the
> > author has some accelerated techniques for proving that things either
> occur
> > or don't.
> >
> > In particular I don't think symmetry is a big problem: you can prove that
> > if a polyomino occurs, it occurs in all orientations; and so if you can
> > prove it doesn't occur in one orientation, you don't have to bother with
> > the others.
> >
> > In the main diagram, I can see the straight tromino but not the L, and
> can
> > easily believe that the L is impossible.
> >
> > I can see the T tetromino, and no others.
> >
> > The author claims 3 pentominoes are possible; I can only see the straight
> > pentomino (pentane, [11111]), and the "V" (1-ethylpropane, [117]). I
> don't
> > know what the third claimed pentomino is, but maybe I'm just blind. It's
> > like "Where's Waldo" on steroids.
> >
> >
> > On Wed, Sep 14, 2022 at 2:00 PM Tom Duff <eigenvectors at gmail.com> wrote:
> >
> > > Dunno. I see all 5 tetrominoes in the black squares of the diagram he
> > gives
> > > at https://oeis.org/A347521/a347521.pdf
> > > (Btw, I recomputed the diagram to be sure that his wasn't busted
> > somehow.)
> > > Some of them I just see as sub-polyominoes of larger polyominoes
> (without
> > > looking very carefully).
> > > Maybe he means to divide the diagram into maximal polyominoes and maybe
> > > there are some that can't occur maximally.
> > > How would you do that? The diagram is infinite, so you can't proceed by
> > > inspection.
> > > Proving that a particular polyomino can't occur at any position and in
> > any
> > > orientation with all coordinates coprime, and none of the
> > > orthogonally adjacent cells coprime seems like it would be a
> Diophantine
> > > (Euclidean?) nightmare. Or I'm just not sufficiently clever. (It's
> > happened
> > > before...)
> > >
> > > In any case, A347521 is not sufficiently descriptive.
> > >
> > > On Wed, Sep 14, 2022 at 5:28 AM John Mason <masonmilan33 at gmail.com>
> > wrote:
> > >
> > > > Hi sequence fans
> > > > I have looked at this sequence but I don't know how it works.
> > > > Could anyone maybe give an explanation of, e.g., a(4)=1, say by
> > > > illustrating why 4 of the 5 tetrominoes are excluded and just 1 is
> > > > included?
> > > > I would contact the author but his wiki page says no email address is
> > > > available.
> > > > Thanks
> > > > john
> > > >
> > > > --
> > > > Seqfan Mailing list - http://list.seqfan.eu/
> > > >
> > >
> > > --
> > > Seqfan Mailing list - http://list.seqfan.eu/
> > >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>