A160170 X-toothpick sequence on Z^3 lattice (see Comments for precise definition).

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

Offset: 0

Author

Omar E. Pol, May 03 2009

Keywords

• nonn

Comments

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.

Links

• 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)
• Olbaid Fractalium, Toothpick sequence Part 1, (Watch from minute 3:14), Youtube video (2023).
• 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

Extensions

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