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Spiral tiling numbers.
0

%I #17 Feb 01 2022 07:11:13

%S 3,4,6,12,36,42,48,192,294,324,768,2058,2916,3072,12288,14406,26244,

%T 49152,100842,196608,236196,705894,786432,2125764,3145728,4941258,

%U 12582912,19131876,34588806,50331648,172186884,201326592,242121642,805306368,1549681956

%N Spiral tiling numbers.

%C This is basically three intertwined sequences:

%C triangular: 3*1, 3*4, 3*16, ... 3*4^n;

%C square: 4*1, 4*9, 4*81, ... 4*9^n;

%C hexagonal: 6*1, 6*7, 6*49, ... 6*7^n.

%C Each number in the above sequence has a particular geometric interpretation:

%C 3: a single triangular tile

%C 4: a single square tile

%C 6: a single hexagonal tile

%C 12 = 3*4 = triangle (three sides) * 4 tiles = one triangle in the center, with 3 equally sized triangles surrounding it;

%C 36 = 4*9 = square (four sides) * 9 tiles = one square with 8 other similar copies surrounding it;

%C 42 = 6*7 = hexagon (six sides) * 7 tiles = one hexagon with 6 other copies surrounding it.

%C Each number in the sequence has a prime factorization which uniquely determines whether it corresponds to a triangular, square or hexagonal tiling and the tiling's size. Factorization and rewriting into canonical form effectively becomes the inverse operation of "mixing" the three initial sequences.

%H Paul Bourke, <a href="http://paulbourke.net/texture_colour/tilingplane/3.gif">Spiral Honeycomb Mosaic of order 3</a>

%e 2058 = 6 * 7^3 = spiral honeycomb (hexagonal) mosaic of order 3, which is depicted in the linked image.

%t With[{nn=40},Take[Union[Flatten[Table[{3*4^Range[0,n], 4*9^Range[0,n], 6*7^Range[ 0,n]},{n,nn}]]],nn]] (* _Harvey P. Dale_, Oct 20 2012 *)

%Y Cf. A003401.

%K nonn,uned

%O 1,1

%A Declan Malone (declan.malone+sloane(AT)gmail.com), Apr 20 2008

%E More terms from _Harvey P. Dale_, Oct 20 2012