[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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