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A137333
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Spiral tiling numbers.
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0
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3, 4, 6, 12, 36, 42, 48, 192, 294, 324, 768, 2058, 2916, 3072, 12288, 14406, 26244, 49152, 100842, 196608, 236196, 705894, 786432, 2125764, 3145728, 4941258, 12582912, 19131876, 34588806, 50331648, 172186884, 201326592, 242121642, 805306368, 1549681956
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OFFSET
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1,1
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COMMENTS
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This is basically three intertwined sequences:
triangular: 3*1, 3*4, 3*16, ... 3*4^n;
square: 4*1, 4*9, 4*81, ... 4*9^n;
hexagonal: 6*1, 6*7, 6*49, ... 6*7^n.
Each number in the above sequence has a particular geometric interpretation:
3: a single triangular tile
4: a single square tile
6: a single hexagonal tile
12 = 3*4 = triangle (three sides) * 4 tiles = one triangle in the center, with 3 equally sized triangles surrounding it;
36 = 4*9 = square (four sides) * 9 tiles = one square with 8 other similar copies surrounding it;
42 = 6*7 = hexagon (six sides) * 7 tiles = one hexagon with 6 other copies surrounding it.
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.
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LINKS
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EXAMPLE
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2058 = 6 * 7^3 = spiral honeycomb (hexagonal) mosaic of order 3, which is depicted in the linked image.
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,uned
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AUTHOR
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Declan Malone (declan.malone+sloane(AT)gmail.com), Apr 20 2008
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EXTENSIONS
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STATUS
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approved
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