

A160170


Xtoothpick sequence on Z^3 lattice (see Comments for precise definition).


14



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
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OFFSET

0,3


COMMENTS

Here a "Xtoothpick" is defined to be a cross with 4 endpoints and a midpoint. Also, a Xtoothpick can be represented by set of four connected toothpicks forming a cross.
We start at stage 0 on the Z^3 lattice with no Xtoothpicks.
At stage 1 place a Xtoothpick.
Rule: each exposed endpoint of the Xtoothpicks of the old generation must be touched by the midpoint of a Xtoothpick of new generation (see illustrations).
The sequence gives the number of Xtoothpicks in the threedimensional structure after nth stage. A170171 (the first differences) gives the number of Xtoothpicks added at nth stage.
For a similar sequence but starting with a single toothpick see A170876.
For the Xtoothpick sequence on Z^2 lattice see A147562, the UlamWarburton cellular automaton.
For more information about the growth of toothpicks see A139250.


LINKS

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), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)1) for n >= 2.],
R. J. Mathar, C++ program
R. J. Mathar, Views after stages 1 to 10 (PDF)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences


CROSSREFS

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


KEYWORD

nonn


AUTHOR

Omar E. Pol, May 03 2009, Dec 13 2010, Mar 28 2011


EXTENSIONS

C++ program, illustrations and more terms (a(6)a(38)) based on Email from R. J. Mathar dated Jan 10 2010.


STATUS

approved



