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A160170 X-toothpick 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 (list; graph; refs; listen; history; text; internal format)



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.


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, 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


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




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


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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Last modified March 7 14:53 EST 2021. Contains 341897 sequences. (Running on oeis4.)