[seqfan] Re: Help with A028247 Number of T-frame polyominoes with n cells

[Allan Wechsler]

Three points about A028247:

1. I am glad we have been able to puzzle out the original intent for this
sequence. Let's make it easier for the next generation, and add some
clarification in the comments section. We now have two or three equivalent
formulations, any of which will suffice.

2. I wonder whether A028247 ought to be extended backward to note that
there are 0 examples for n=1, 2, and 3. We usually do this in similar
cases; see, for example, https://oeis.org/A006748.

3. Does a similar sequence to A028247 exist for "L-frame" polyominoes?
These would be those consisting of a rectangle with a single rectangular
notch at one corner. Equivalently, those with a right-angled hexagonal
perimeter, with turns LLLLR. I think, for n starting at 1, the numbers
would be 0,0,1,1,3,3.5,7,8; I can only find one match, and it's not very
plausible. (I'm not super-sure of the later numbers, though.)

On Wed, Feb 1, 2023 at 11:59 AM John Mason <masonmilan33 at gmail.com> wrote:

> Allan
> Thanks, that's right. I changed my test to:
> convex, with a complete top row (extends over the entire width), and a
> bottom row that starts a bit to the right of the top row and finishes a bit
> to the left, and a perimeter that forms a self-avoiding polygon of bracelet
> length 8.
> Now I get the same figures.
>
> Don. For a(8) I think you're missing
>
> OOOO
>
> OOO_
> __O_
>
> On Wed, Feb 1, 2023 at 4:16 PM Allan Wechsler <acwacw at gmail.com> wrote:
>
> > John Mason, your #4 doesn't satisfy what I infer to be Anne Fontaine's
> > intended constraints.
> >
> > One way to say it is that the perimeter must be a right angled octagon,
> and
> > as you traverse it counterclockwise you encounter turns in the order
> > LLLLRLLR.
> >
> > Fontaine is a combinatorial tiling specialist, and has written about
> > "U-frame" (LLLLLLRR) and "Z-frame" (LLLRLLLR) polyominoes as well.
> >
> > On Wed, Feb 1, 2023, 4:45 AM John Mason <masonmilan33 at gmail.com> wrote:
> >
> > > Thanks for everyone's help. That got me through to a(6).
> > > But now, instead of a(7)=10, I'm getting 11. Which of these is wrong?
> > > (I am using the following condition: the polyomino must be board-pile,
> > > convex, have a complete top row (extends over the entire width), and a
> > > bottom row that starts a bit to the right of the top row and finishes a
> > bit
> > > to the left)
> > >
> > > 1.
> > > OOO
> > > OOO
> > > _O_
> > >
> > > 2.
> > > OOOOO
> > > _OO__
> > >
> > > 3.
> > > OOOO
> > > OO__
> > > _O__
> > >
> > > 4.
> > > OOOO
> > > _OO_
> > > _O__
> > >
> > > 5.
> > > OOO
> > > OO_
> > > _O_
> > > _O_
> > >
> > > 6.
> > > OOOOOO
> > > _O____
> > >
> > > 7.
> > > OOOOOO
> > > __O___
> > >
> > > 8.
> > > OOOOO
> > > _O___
> > > _O___
> > >
> > > 9.
> > > OOOOO
> > > __O__
> > > __O__
> > >
> > > 10.
> > > OOOO
> > > _O__
> > > _O__
> > > _O__
> > >
> > > 11.
> > > OOO
> > > _O_
> > > _O_
> > > _O_
> > > _O_
> > >
> > > On Wed, Feb 1, 2023 at 5:38 AM Don Reble via SeqFan <
> > seqfan at list.seqfan.eu
> > > >
> > > wrote:
> > >
> > > > > XXX
> > > > > XX.
> > > > > .X.
> > > >
> > > > > ...polyominoes which are integral rectangles with integral notches
> > cut
> > > > > from two adjacent corners.
> > > >
> > > >     I like that take, yielding sequence
> > > >
> >  0,0,0,1,2,6,10,18,27,42,55,80,101,133,163,211,246,308,353,432,...
> > > >     but I get a(8)=18 instead of 19, etc. Did I miss one?
> > > >
> > > > .XXX. .XX... ..XX.. .X..... ..X.... ...X...
> > > > XXXXX XXXXXX XXXXXX XXXXXXX XXXXXXX XXXXXXX
> > > >
> > > > .XX. .X... .X.... ..X...
> > > > .XX. XX... .X.... ..X...
> > > > XXXX XXXXX XXXXXX XXXXXX
> > > >
> > > > .X. .X. .X.. .X... ..X..
> > > > XX. .X. .X.. .X... ..X..
> > > > XX. XXX XX.. .X... ..X..
> > > > XXX XXX XXXX XXXXX XXXXX
> > > >
> > > > .X. .X..
> > > > .X. .X..
> > > > .X. .X..
> > > > XX. .X..
> > > > XXX XXXX
> > > >
> > > > .X.
> > > > .X.
> > > > .X.
> > > > .X.
> > > > .X.
> > > > XXX
> > > >
> > > > --
> > > > Don Reble  djr at nk.ca
> > > >
> > > >
> > > > --
> > > > 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/
>