New Polyform Sequence Idea

[Rhys F]

Hello everyone. I have a new idea for a polyform sequence. I call them 'polyiamondominoes', since they are combinations of polyiamonds and polyominoes. They are polyforms made out of triangles and squares. In this case, specifically, the number of triangles and squares are the same. We are counting free polyforms, and the squares/triangles cannot overlap. Triangles and squares can connect in any way, but must be via an edge. They are not a subset of a triangle and square tiling.

I did some manual calculations and found that a(1) = 1 and a(2) = 15. I call them 'Moniamondmonominoes' and 'Diamondominoes'. (Yes, these names are long; perhaps we could shorten them?) You can see them here: https://rhys-feltman-polyforms.neocities.org/polyiamondominoes

I have a hypothesis that a(n) = A000105(2n)*A000577(2n), which works for the first 2 terms, and estimates a(3) at 420. However, I started counting manually, but halfway through I stopped due to the amount of work to count all of them. The incomplete result was 232, which is way below this estimate.

I would appreciate help in finding a(3). I'm not sure how to do it another way, but maybe one of you knows how to code this? If you're interested, please let me know. Thank you.

[Pontus von Brömssen]

Without the requirement that there must be equally many triangles and squares, this is A390142. There's a linked C++ program that might be useful.

/Pontus 

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[Rhys F]

Is the script in another program? I only have Python and PARI. I downloaded C++, but I'm not sure how to use it.

[Rhys F]

With the help of a Python script, I have determined a(3) = 409, which is remarkably close to the 420 estimate earlier, made by a(n) ~ A000105(2n) * A000577(2n). My script took seconds to verify a(1) and a(2), and 2 minutes to find a(3), so I'm sure a(4) will take much longer. I currently already have 3 drafts open, so once one of them is published, I will make this a new draft.

[Arthur O'Dwyer]

On Fri, Apr 10, 2026 at 6:38 PM Rhys F <feltma...@gmail.com> wrote:

Hello everyone. I have a new idea for a polyform sequence. I call them 'polyiamondominoes', since they are combinations of polyiamonds and polyominoes. They are polyforms made out of triangles and squares. In this case, specifically, the number of triangles and squares are the same.

FWIW, I don't think your naming scheme is quite right. Normally a "polyfoo" is what you get by gluing together N foos. In your case, the units you're gluing together aren't actually units! You always add a square and a triangle at the same time, but they don't have to be placed "together." Also, this means your numbers grow rapidly in an unwieldy manner — it grows exponentially as with all polyforms, but twice as exponentially, because you're adding two pieces at each step.

(1)

One way to make this a more traditional polyform is to look instead at polyhouses, where I'll coin the term "house" for the conglomeration of a triangle and a square. These are the dihouses among your original set:

I don't see this particular polyform mentioned on either

https://web.archive.org/web/20251122230154/http://www.vicher.cz/puzzle/polyforms.htm

https://web.archive.org/web/20030803065351/http://members.aol.com/wgreview/polyenum.html

That's probably because these polyforms are aggressively non-space-filling: you obviously can't tile the plane with them in any interesting way. That makes them less recreationally interesting IMHO (but not totally uninteresting). Incidentally, Kate Jones' Tri-Chex tiles the plane with a proper subset of (not the polyhouses but) the guys below.

Note that the "benzene ring" hexahouse is not chiral.

(2)

A different way to make this a more traditional polyform but also keep the numbers from growing so rapidly would be to look at polyforms where each unit would be "either a triangle or a square but I don't care which." That sequence could be represented as a 2D table: s=number of squares, t=number of triangles.

    s=0  1  2  3  4  5

--------------------- 

t=0 |    1  1  2  5 12

t=1 | 1  1  3 11

t=2 | 1  3 15

t=3 | 1  7

t=4 | 3

t=5 | 4

 

Here the first row (t=0) is A000105. The first column (s=0) is A000577. The diagonal (1, 15, ...) is your original sequence. The antidiagonal sums (2, 3, 9, 41, ...) are A390142!

I don't have a great name for this kind of thing, though. I want to say "polyindifferents," except that that single word fails to express that we're specifically indifferent between equilateral triangle and square. There would be other tables for other species of polyindifferents (e.g. indifferent between hex and triangle, indifferent between square and tan (A390999!), indifferent between triangle and drafter, indifferent between monomino and domino...

All these "polyindifferents" suffer from the infelicity that two polyindifferents of the same order can have different areas. (A triangle unit doesn't have the same area as a square unit.) This can mean that the same shape is both an n-indifferent and an (n+1)-indifferent. Notice the unit square is counted in both the first and second rows of A390999's image, first as a monoindifferent (square) and then as a di-indifferent (two tans joined at the hypotenuse). That feels infelicitous to me, in general.

To keep the notion of indifference but restore the felicitous equal-area invariant, you could look for example at polytrominoes — polyindifferents where the two indifferent units are the I-tromino and the L-tromino. I recently looked at poly-I-trominoes ("polybars of order 3"); see here.

my $.02,

–Arthur

[John Mason]

Hi

With a program that generates these polyforms (without the equality condition) consistently with A390142 up to size 6, I am getting the following counts for size 6 polyforms with 0 squares, 1 square, 2 squares, ...

0	12
1	71
2	229
3	385
4	388
5	177
6	35

So my estimate for this question would be 385.

However, I must admit that my program is currently giving results for size 7 that are different to A390142, so look at these numbers cum grano salis.

You can see a rather amateurish rendering at : https://drive.google.com/file/d/1pZaUG4yib_NV-b9GYlC4WvG5PuyqAuoE/view?usp=sharing

john

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[Chris Scussel]

Hello all,

    Another variation is to strictly alternate adding triangles and squares.  This forms two sequences: one starting with a triangle and the other starting with a square. Even numbered terms of these two sequences are identical, since the numbers of triangles and squares are the same. These sequences grow more slowly than the original sequence and may be easier to investigate.

Best regards,

    Chris

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