A161644 Number of ON states after n generations of cellular automaton based on triangles.

0, 1, 4, 10, 16, 22, 34, 52, 64, 70, 82, 106, 136, 160, 190, 232, 256, 262, 274, 298, 328, 358, 400, 466, 532, 568, 598, 658, 742, 814, 892, 988, 1036, 1042, 1054, 1078, 1108, 1138, 1180, 1246, 1312, 1354, 1396, 1474, 1588, 1702, 1816, 1966, 2104, 2164, 2194

Offset: 0

Author

David Applegate and N. J. A. Sloane, Jun 15 2009

Keywords

• nonn

Comments

Analog of A151723 and A151725, but here we are working on the hexagonal net where each triangular cell has three neighbors (meeting along its edges). A cell is turned ON if exactly one of its three neighbors is ON. An ON cell remains ON forever.

We start with a single ON cell.

There is a dual version where the triangular cells meet vertex-to-vertex. The counts are the same: the two versions are isomorphic. Reed (1974) uses the vertex-to-vertex version. See the two Sloane "Illustration" links below to compare the two versions.

It appears that a(n) is also the number of polytoothpicks added in a toothpick structure formed by V-toothpicks but starting with a Y-toothpick: a(n) = a(n-1)+(A182632(n)-A182632(n-1))/2. (Checked up to n=39.) - Omar E. Pol, Dec 07 2010 and R. J. Mathar, Dec 17 2010

It appears that the behavior is similar to A161206. - Omar E. Pol, Jan 15 2016

It would be nice to have a formula or recurrence.

If new triangles are required to always move outwards we get A295559 and A295560.

From Paul Cousin, May 23 2025: (Start)

This is ETA rule 242 (11110010 in binary):

-----------------------------------------------

|state of the cell |1|1|1|1|0|0|0|0|

|sum of the neighbors' states |3|2|1|0|3|2|1|0|

|cell's next state |1|1|1|1|0|0|1|0|

----------------------------------------------- (End)

References

• R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Describes the dual structure where new triangles are joined at vertices rather than edges.]
• S. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962. See Example 3.

Links

• Paul Cousin, Table of n, a(n) for n = 0..16385 (first 10001 terms from Rémy Sigrist)
• David Applegate, The movie version
• 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.]
• Paul Cousin, ETA rule 242
• Paul Cousin, Triangular Automata Integer Sequences
• Lucas Garron, first 64 steps
• Lucas Garron, after 128 steps
• R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]
• Rémy Sigrist, PARI program for A161644
• N. J. A. Sloane, Illustration of first 7 generations of A161644 and A295560 (edge-to-edge version)
• N. J. A. Sloane, Illustration of first 11 generations of A161644 and A295560 (vertex-to-vertex version) [Include the 6 cells marked x to get A161644(11), exclude them to get A295560(11).]
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Crossrefs

Cf. A151723, A151725, A147562, A161206, A161645, A139250, A160120, A161206, A182632, A182840, A250300, A295559, A295560.

Programs

PARI

\\ See Links section.

Formula

a(n) = (A182632(n) - 1)/2, n >= 1. - Omar E. Pol, Mar 07 2013

Extensions

Edited by N. J. A. Sloane, Jan 10 2010 and Nov 27 2017