cp's OEIS Frontend

A160412 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

%I A160412 #21 Nov 02 2022 07:45:21
%S A160412 0,3,12,21,48,57,84,111,192,201,228,255,336,363,444,525,768,777,804,
%T A160412 831,912,939,1020,1101,1344,1371,1452,1533,1776,1857,2100,2343,3072,
%U A160412 3081,3108,3135,3216,3243,3324,3405,3648,3675,3756,3837,4080,4161,4404,4647
%N A160412 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).
%C A160412 From _Omar E. Pol_, Nov 10 2009: (Start)
%C A160412 On the infinite square grid, consider the outside corner of an infinite square.
%C A160412 We start at round 0 with all cells in the OFF state.
%C A160412 The rule: A cell in turned ON iff exactly one of its four vertices is a corner vertex of the set of ON cells. So in each generation every exposed vertex turns on three new cells.
%C A160412 At round 1, we turn ON three cells around the corner of the infinite square, forming a concave-convex hexagon with three exposed vertices.
%C A160412 At round 2, we turn ON nine cells around the hexagon.
%C A160412 At round 3, we turn ON nine other cells. Three cells around of every corner of the hexagon.
%C A160412 And so on.
%C A160412 Shows a fractal-like behavior similar to the toothpick sequence A153006.
%C A160412 For the first differences see the entry A162349.
%C A160412 For more information see A160410, which is the main entry for this sequence.
%C A160412 (End)
%H A160412 Michael De Vlieger, <a href="/A160412/b160412.txt">Table of n, a(n) for n = 0..1000</a>
%H A160412 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 A160412 Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 31.
%H A160412 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polca027.jpg">Illustration of initial terms</a> [From _Omar E. Pol_, Nov 10 2009]
%H A160412 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>
%H A160412 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a> - _Omar E. Pol_, Nov 10 2009
%F A160412 From _Omar E. Pol_, Nov 10 2009: (Start)
%F A160412 a(n) = A160410(n)*3/4.
%F A160412 a(0) = 0, a(n) = A130665(n-1)*3, for n>0.
%F A160412 (End)
%e A160412 If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
%e A160412 ...77..77..77..77
%e A160412 ...766667..766667
%e A160412 ....6556....6556.
%e A160412 ....654444444456.
%e A160412 ...76643344334667
%e A160412 ...77.43222234.77
%e A160412 ......44211244...
%e A160412 00000000001244...
%e A160412 00000000002234.77
%e A160412 00000000004334667
%e A160412 0000000000444456.
%e A160412 0000000000..6556.
%e A160412 0000000000.766667
%e A160412 0000000000.77..77
%e A160412 0000000000.......
%e A160412 0000000000.......
%e A160412 0000000000.......
%t A160412 a[n_] := 3*Sum[3^DigitCount[k, 2, 1], {k, 0, n - 1}]; Array[a, 48, 0] (* _Michael De Vlieger_, Nov 01 2022 *)
%Y A160412 Cf. A139250, A139251, A153006, A152980, A160410, A160414.
%Y A160412 Cf. A130665, A162349. - _Omar E. Pol_, Nov 10 2009
%K A160412 nonn
%O A160412 0,2
%A A160412 _Omar E. Pol_, May 20 2009, Jun 01 2009
%E A160412 More terms from _Omar E. Pol_, Nov 10 2009
%E A160412 Edited by _Omar E. Pol_, Nov 11 2009
%E A160412 More terms from _Nathaniel Johnston_, Nov 06 2010
%E A160412 More terms from _Colin Barker_, Apr 19 2015