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A173530 Number of ON cells after n generations of three-dimensional cellular automaton related to Sierpinski's triangle and the toothpick sequences (See Comments for definition) 3
0, 1, 3, 7, 11, 15, 23, 39, 47, 51, 59, 75, 91, 107, 139, 203, 219, 223, 231, 247, 263, 279, 311, 375, 407, 423, 455, 519, 583, 647, 775, 1031, 1063, 1067, 1075, 1091, 1107, 1123, 1155, 1219, 1251, 1267, 1299, 1363, 1427, 1491, 1619 (list; graph; refs; listen; history; text; internal format)
The structure is similar to Sierpinski's triangle but in this case we are in 3-D.
The triangles of the new generation are arranged on planes that are orthogonal with respect to the planes of the previous generation.
If n is odd then the triangles are arranged on planes that are parallel to the plane XZ.
If n is even then the triangles are arranged on planes that are parallel to the plane YZ.
The sequence A173531 (The first differences) gives the number of triangles added at the n-th stage.
We start with no triangles.
At round 1 we place a triangle anywhere in the space on the plane XZ.
At round 2 we place two other triangles on planes that are parallel to the plane YZ.
At round 3 we place four other triangles on planes that are parallel to the plane XZ.
And so on...
It appears that the three-dimensional pattern has a recursive, fractal (or fractal-like) structure. An animation can show the fractal (or fractal-like) behavior.
Note that the triangles can be replaced by V-toothpicks or L-toothpicks. More generally, the triangles can be replaced by any polytoothpick formed by two toothpicks connected by one of its vertices, with an angle greater than zero degrees and less than 180 degrees.
In this structure every polytoothpick has two components, so after n stages the structure has 2 * a (n) components.
Note that for n <= 11, in all cases (using triangles or polytoothpicks), one of the views of the 3-D structure is equal to the toothpick structure of A139250 (See illustrations).
See the entries A139250, A161206 and A172310 for more information about the growth of toothpicks, V-toothpicks and L-toothpicks.
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Partial sums of A173531.
Sequence in context: A266535 A182634 A365142 * A192114 A145052 A100900
Omar E. Pol, Oct 10 2010

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