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A160170
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X-toothpick sequence on Z^3 lattice (see Comments for precise definition).
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14
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0, 1, 5, 13, 21, 45, 77, 109, 165, 245, 325, 413, 525, 685, 853, 1093, 1317, 1661, 1981, 2301, 2645, 3093, 3621, 4157, 4861, 5565, 6437, 7173, 8053, 8893, 9917, 11005, 12261, 13589, 14981, 16397, 17837, 19341, 20997
(list;
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listen;
history;
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OFFSET
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0,3
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COMMENTS
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Here a "X-toothpick" is defined to be a cross with 4 endpoints and a midpoint. Also, a X-toothpick can be represented by set of four connected toothpicks forming a cross.
We start at stage 0 on the Z^3 lattice with no X-toothpicks.
At stage 1 place a X-toothpick.
Rule: each exposed endpoint of the X-toothpicks of the old generation must be touched by the midpoint of a X-toothpick of new generation (see illustrations).
The sequence gives the number of X-toothpicks in the three-dimensional structure after n-th stage. A170171 (the first differences) gives the number of X-toothpicks added at n-th stage.
For a similar sequence but starting with a single toothpick see A170876.
For the X-toothpick sequence on Z^2 lattice see A147562, the Ulam-Warburton cellular automaton.
For more information about the growth of toothpicks see A139250.
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LINKS
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Table of n, a(n) for n=0..38.
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.],
R. J. Mathar, C++ program
R. J. Mathar, View after stage 1
R. J. Mathar, View after stage 2
R. J. Mathar, View after stage 3
R. J. Mathar, View after stage 4
R. J. Mathar, View after stage 5
R. J. Mathar, View after stage 6
R. J. Mathar, View after stage 7
R. J. Mathar, View after stage 8
R. J. Mathar, View after stage 9
R. J. Mathar, View after stage 10
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences
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CROSSREFS
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Cf. A139250, A160120, A160160, A160171, A170876.
Cf. A147562. - Omar E. Pol, Mar 28 2011
Sequence in context: A273750 A191116 A166051 * A277189 A060004 A174172
Adjacent sequences: A160167 A160168 A160169 * A160171 A160172 A160173
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KEYWORD
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nonn
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AUTHOR
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Omar E. Pol, May 03 2009, Dec 13 2010, Mar 28 2011
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EXTENSIONS
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C++ program, illustrations and more terms (a(6)-a(38)) based on Email from R. J. Mathar dated Jan 10 2010.
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STATUS
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approved
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