

A137333


Spiral tiling numbers.


0



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

1,1


COMMENTS

This is basically three intertwined sequences:
triangular: 3 x 1, 3 x 4, 3 x 16, ... 3 x 4 ^ n
square: 4 x 1, 4 x 9, 4 x 81, ... 4 x 9 ^ n
hexagonal: 6 x 1, 6 x 7, 6 x 49, ... 6 x 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 x 4 = triangle (three sides) x 4 tiles = one triangle in the center, with 3 equallysized triangles surrounding it
36 = 4 x 9 = square (four sides) x 9 tiles = one square with 8 other similar copies surrounding it
42 = 6 x 7 = hexagon (six sides) x 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.


LINKS

Table of n, a(n) for n=1..35.
Paul Bourke, Spiral Honeycomb Mosaic of order 3


EXAMPLE

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


MATHEMATICA

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 *)


CROSSREFS

Cf. A003401.
Sequence in context: A160684 A176045 A255733 * A006719 A202855 A182857
Adjacent sequences: A137330 A137331 A137332 * A137334 A137335 A137336


KEYWORD

nonn,uned


AUTHOR

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


EXTENSIONS

More terms from Harvey P. Dale, Oct 20 2012


STATUS

approved



