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A267190 Number of ON cells after n generations of the cellular automaton on the square grid that is described in the Comments.

%I A267190 #51 Feb 24 2021 02:48:19
%S A267190 0,1,5,9,13,25,29,41,53,65,85,97,117,145,149,161,173,193,221,241,277,
%T A267190 313,357,401,437,489,541,553,581,609,645,689,733,801,869,945,1021,
%U A267190 1081,1149,1217,1277,1345,1397,1433,1501,1569,1653,1753,1829,1905,1997,2057,2141,2225,2317,2449,2549,2681,2797,2889,2965,3041,3149,3289
%N A267190 Number of ON cells after n generations of the cellular automaton on the square grid that is described in the Comments.
%C A267190 The cells are the squares of the standard square grid.
%C A267190 Cells are either OFF or ON, once they are ON they stay ON, and we begin in generation 1 with 1 ON cell.
%C A267190 Each cell has 4 neighbors, those that it shares an edge with. Cells that are ON at generation n all try simultaneously to turn ON all their neighbors that are OFF. They can only do this at this point in time; afterwards they go to sleep (but stay ON).
%C A267190 A square Q is turned ON at generation n+1 if:
%C A267190 a) Q shares an edge with one and only one square P (say) that was turned ON at generation n (in which case the two squares which intersect Q only in a vertex not on that edge are called Q's “outer squares”), and
%C A267190 b) Q's outer squares were not turned ON in any previous generation, and
%C A267190 c) Q's outer squares are not prospective squares of the (n+1)st generation satisfying a).
%C A267190 A151895, A151906, and A170896 are closely related cellular automata.
%C A267190 The key difference between this and A170896 is that if we have two squares Q1 and Q2, both satisfying a), and that are each an outer square of the other, where Q1 satisfies b), but Q2 does not, then for A170896 Q1 is accepted, but for this sequence Q1 is eliminated.  This first happens at n=14, when, for example, A170896 turns (8,3) ON but A267190 doesn't (because (9,2) fails to satisfy b) because (8,1) is ON). - _David Applegate_, Jan 30 2016
%C A267190 A151895 and A267190 first differ at n=17, when A267190 turns (12,2) ON even though its outer square (11,1) was considered (not turned ON) in a previous generation. - _David Applegate_, Jan 30 2016
%D A267190 D. 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
%H A267190 David Applegate, <a href="/A267190/b267190.txt">Table of n, a(n) for n = 0..260</a>
%H A267190 David Applegate, <a href="/A139250/a139250.anim.html">The movie version</a>
%H A267190 David Applegate, <a href="/A267190/a267190_2.pdf">After 16 generations, illustrating a(16)=173 (with the A267191(16)=12 newly created cells shown in blue)</a>
%H A267190 David Applegate, <a href="/A267190/a267190_3.pdf">After 36 generations, illustrating a(36)=1021 (with the A267191(36)=76 newly created cells shown in blue)</a> 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.], which is also available at <a href="http://arxiv.org/abs/1004.3036">arXiv:1004.3036v2</a>
%H A267190 S. M. Ulam, <a href="/A002858/a002858.pdf">On some mathematical problems connected with patterns of growth of figures</a>, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 [Annotated scanned copy]
%H A267190 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%H A267190 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A267190 We do not know of a recurrence or generating function.
%Y A267190 Cf. A267191 (first differences), A151895, A151906, A170896.
%Y A267190 See also A139250.
%K A267190 nonn
%O A267190 0,3
%A A267190 _David Applegate_ and _N. J. A. Sloane_, Jan 21 2016
%E A267190 Corrected by _David Applegate_, Jan 30 2016