[seqfan] Counting ells.

Allan Wechsler acwacw at gmail.com
Sun May 22 20:48:14 CEST 2022

I'm trying to count the number of integral ells of with N cells. An ell is
a rectangle with a rectangular notch cut out of one corner; another way to
say this is that an ell is a polyomino whose boundary is a hexagon.

You can't arrange 1 or 2 cells into an ell, but 3 cells can be arranged
exactly one way.

Now, this is an example of a sequence that I was certain would be in OEIS
already. It's a simple sort of number-of-representations-of-the-form idea,
of which we have many hundreds of examples at least. If you think of this
as a sort of riff on https:/oeis.com/A038548, which is the number of
rectangular polyominoes with N cells, this idea becomes even more obvious.

But my data doesn't match anything in OEIS. For N ranging from 1 to 9, my
hand calculations yield


which is not in OEIS anywhere.

What is *most likely* is that I have blundered and one or more of my nine
numbers is wrong. After all, I'm counting by hand, and I may have missed
something or counted it twice.

It would be great if other seqfans could either confirm or correct my data.

There are several other ways to give the definition. It's the number of
representations of N in the form ab - cd where c < a  and d < b, and a <=
b. Or the number of partitions of N with exactly two part sizes, where the
largest part does not exceed the number of parts. At any rate, I'm
surprised to have come up blank, and bet I have miscounted something.

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