[seqfan] Re: Integers tiling the plane

[Eric Angelini]

Hello SeqFans,
Lars Blomberg has computed the 109 integers < one million which 
tile the plane (as defined here, in French -- and below, in English):
http://www.cetteadressecomportecinquantesignes.com/Paveur.htm

P = 9, 99, 366, 636, 663, 999, 3777, 7377, 7737, 7773, 9999, 17777, 33555, 34554, 35355, 35535, 35553, 43455, 45543, 53355, 53535, 53553, 54345, 55335, 55353, 55434, 55533, 71777, 77177, 77717, 77771, 99999, 132323, 153535, 231323, 232313, 243434, 313232, 323132, 323231, 336666, 342434, 343424, 351535, 353515, 354545, 366366, 366663, 424343, 434243, 434342, 453545, 454535, 515353, 535153, 535351, 535454, 545354, 545453, 633666, 636636, 663366, 663663, 666336, 666633, 999999, 1255255, 1525255, 1535535, 1552525, 1552552, 1553355, 2155255, 2515525, 2525155, 2525515, 2551255, 2551525, 2552155, 2552551, 3355155, 3515355, 3551553, 3553515, 5125525, 5152525, 5153553, 5155252, 5155335, 5215525, 5251552, 5252515, 5252551, 5255125, 5255152, 5255215, 5335515, 5351535, 5355351, 5512552, 5515252, 5515533, 5521552, 5525251, 5525512, 5525521, 5533551, 5535153, 9999999, ...

Congratulation to Lars!
The problem of counting tiled "rectangles" of size m x n and "boxes" of size a x b x c is still open...
Best,
É.

----------

> Take an infinite plane tiled by initially empty squares of size 1 x 1;
> Law1: such a square can only be filled by a one-digit integer d;
> Law2: we call "neighborhood" of d the 3 x 3 square centered on d;
> Law3: if d occupies a square, there must be in the neighborhood of d
  exactly d copies of d.
(...)