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

[John Mason]

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
> > >
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> > >
> >
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>
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