cp's OEIS Frontend

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

%I A161644 #99 May 27 2025 01:05:51
%S A161644 0,1,4,10,16,22,34,52,64,70,82,106,136,160,190,232,256,262,274,298,
%T A161644 328,358,400,466,532,568,598,658,742,814,892,988,1036,1042,1054,1078,
%U A161644 1108,1138,1180,1246,1312,1354,1396,1474,1588,1702,1816,1966,2104,2164,2194
%N A161644 Number of ON states after n generations of cellular automaton based on triangles.
%C A161644 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.
%C A161644 We start with a single ON cell.
%C A161644 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.
%C A161644 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
%C A161644 It appears that the behavior is similar to A161206. - _Omar E. Pol_, Jan 15 2016
%C A161644 It would be nice to have a formula or recurrence.
%C A161644 If new triangles are required to always move outwards we get A295559 and A295560.
%C A161644 From _Paul Cousin_, May 23 2025: (Start)
%C A161644 This is ETA rule 242 (11110010 in binary):
%C A161644   -----------------------------------------------
%C A161644   |state of the cell            |1|1|1|1|0|0|0|0|
%C A161644   |sum of the neighbors' states |3|2|1|0|3|2|1|0|
%C A161644   |cell's next state            |1|1|1|1|0|0|1|0|
%C A161644   ----------------------------------------------- (End)
%D A161644 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.]
%D A161644 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.
%H A161644 Paul Cousin, <a href="/A161644/b161644.txt">Table of n, a(n) for n = 0..16385</a> (first 10001 terms from Rémy Sigrist)
%H A161644 David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H A161644 David Applegate, Omar E. Pol and N. J. A. Sloane, <a href="/A000695/a000695_1.pdf">The Toothpick Sequence and Other Sequences from Cellular Automata</a>, 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.]
%H A161644 Paul Cousin, <a href="https://triangular-automata.net/rules.html?rule=242">ETA rule 242</a>
%H A161644 Paul Cousin, <a href="https://triangular-automata.net/?p=integer-sequences">Triangular Automata Integer Sequences</a>
%H A161644 Lucas Garron, <a href="/A161644/a161644.gif">first 64 steps</a>
%H A161644 Lucas Garron, <a href="/A161644/a161644.png">after 128 steps</a>
%H A161644 R. Reed, <a href="/A005448/a005448_1.pdf">The Lemming Simulation Problem</a>, 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]
%H A161644 Rémy Sigrist, <a href="/A161644/a161644.gp.txt">PARI program for A161644</a>
%H A161644 N. J. A. Sloane, <a href="/A161644/a161644_1.png">Illustration of first 7 generations of A161644 and A295560 (edge-to-edge version)</a>
%H A161644 N. J. A. Sloane, <a href="/A161644/a161644_2.png">Illustration of first 11 generations of A161644 and A295560 (vertex-to-vertex version)</a> [Include the 6 cells marked x to get A161644(11), exclude them to get A295560(11).]
%H A161644 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%F A161644 a(n) = (A182632(n) - 1)/2, n >= 1. - _Omar E. Pol_, Mar 07 2013
%o A161644 (PARI) \\ See Links section.
%Y A161644 Cf. A151723, A151725, A147562, A161206, A161645, A139250, A160120, A161206, A182632, A182840, A250300, A295559, A295560.
%K A161644 nonn
%O A161644 0,3
%A A161644 _David Applegate_ and _N. J. A. Sloane_, Jun 15 2009
%E A161644 Edited by _N. J. A. Sloane_, Jan 10 2010 and Nov 27 2017