Self-integers tiling the plane

Ron H. Hardin’s sequence (http://oeis.org/A197049) has given me the idea to tile the plane with integers in such a manner that any digit of the plane would describe the quantity of copies of itself it orthogonally touches.

In the hereunder box, for example, all zeroes have 0 copies of 0 as “ortho-neighbor”, all ones have 1 copy of 1 as neighbor, all twos touch two other “2”:

2 2 2 2 2

2 0 1 1 2

2 2 2 2 2

4 is the smallest integer tiling the plane (as 44 does):

...

4 4 4 4

... 4 4 4 4

4 4 4 4 ...

4 4 4 4

...

Is 233 the next one?

...

2 3 3 2 3 3 2 3 3 2 3 3 2 3 3

2 3 3 2 3 3 2 3 3 2 3 3 2 3 3

... 2 3 3 2 3 3 2 3 3 2 3 3 2 3 3

2 3 3 2 3 3 2 3 3 2 3 3 2 3 3 ...

2 3 3 2 3 3 2 3 3 2 3 3 2 3 3

2 3 3 2 3 3 2 3 3 2 3 3 2 3 3

...

323 and 332 tile the plane too, of course; and 233233, 323323, 332332, etc.

30333 does the job:

3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3

3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0

3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3

0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3

3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3

3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3

3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0

3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3

0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3

3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3

3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3 3 0 3 3 3

120212 is a candidate:

1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2

0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2

1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2

1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2

0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2

1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2

1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2

0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2

1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2

1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2 1 2 0 2 1 2

What would be the sequence of “self-integers” tiling the plane? Does S start like this?

S = 4, 44, 233, 323, 332, 444, ...

Best,

É.

__________

[Remark #1]:

I think the huge integer 1133133333133303331333331331133333333 tiles the infinite plane – is thus part of S – because of the hereunder tile:

3 3 1 3 3 3 0

3 3 1 3 3 3

3 3 3 3 1 1

1 1 3 3 3 3

3 3 3 1 3 3

3 3 3 1 3 3

Make a copy of this tile and stick its left side against its right side – under the zero. Repeat.

The same happens with 133033133 and the tile:

1 3 3 0 3 3 1 3 3

1 3 3 1 3 3 0 3 3

0 3 3 1 3 3 1 3 3

[Remark #2]:

Some infinite tilings are impossible to build with a single integer:

0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0

1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1

1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1

0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0

1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1

1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1

0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0

1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1

1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1

0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0

1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1

1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1

0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0

... or this one:

1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2

2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2

2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1

0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2

1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2

1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0

0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1

1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1

2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0

2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1

1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2

2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2

2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1

0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2

1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2

1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0

0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1

1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1

2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0

2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1 0 2 2 1 2 2 1 0 1 1

1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2 1 2 2 0 1 1 0 1 2 2

... or that:

2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2

2 0 2 1 1 0 1 1 2 0 2 1 1 0 1 1 2 0 2 1 1 0 1 1 2 0 2

2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2

1 0 1 1 2 0 2 1 1 0 1 1 2 0 2 1 1 0 1 1 2 0 2 1 0 1 1

2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2

2 0 2 1 1 0 1 1 2 0 2 1 1 0 1 1 2 0 2 1 1 0 1 1 2 0 2

2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2

1 0 1 1 2 0 2 1 1 0 1 1 2 0 2 1 1 0 1 1 2 0 2 1 0 1 1

2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2

2 0 2 1 1 0 1 1 2 0 2 1 1 0 1 1 2 0 2 1 1 0 1 1 2 0 2

2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2 0 2 2 2

1 0 1 1 2 0 2 1 1 0 1 1 2 0 2 1 1 0 1 1 2 0 2 1 0 1 1

[Remark #3]:

Some integers might tile the plane in more than one way.

Best,

É.