[seqfan] Self-integers tiling the plane

Eric Angelini Eric.Angelini at kntv.be
Mon Oct 10 14:04:36 CEST 2011


Hello SeqFans,

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.

[This message is in full color here:
 http://www.cetteadressecomportecinquantesignes.com/SelfTiles.htm]

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 2 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

Etc.
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.

 

 





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