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A161644 Number of ON states after n generations of cellular automaton based on triangles. 15
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 (list; graph; refs; listen; history; text; internal format)



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.

Needs a b-file.


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.]


Table of n, a(n) for n=0..50.

David Applegate, The movie version

David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata

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]

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


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


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

Sequence in context: A109273 A294636 A295560 * A215032 A184527 A246297

Adjacent sequences:  A161641 A161642 A161643 * A161645 A161646 A161647




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


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



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Last modified February 22 11:16 EST 2018. Contains 299452 sequences. (Running on oeis4.)