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

[Allan Wechsler]

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/
>