cp's OEIS Frontend

A183060 Number of "ON" cells at n-th stage in a simple 2-dimensional cellular automaton (see Comments for precise definition).

%I A183060 #40 Jul 14 2024 20:02:44
%S A183060 0,1,4,7,14,17,24,31,50,53,60,67,86,93,112,131,186,189,196,203,222,
%T A183060 229,248,267,322,329,348,367,422,441,496,551,714,717,724,731,750,757,
%U A183060 776,795,850,857,876,895,950,969,1024,1079,1242,1249,1268,1287
%N A183060 Number of "ON" cells at n-th stage in a simple 2-dimensional cellular automaton (see Comments for precise definition).
%C A183060 On the semi-infinite square grid, start with all cells OFF.
%C A183060 Turn a single cell to the ON state in row 1.
%C A183060 At each subsequent step, each cell with exactly one neighbor ON is turned ON, and everything that is already ON remains ON.
%C A183060 The sequence gives the number of "ON" cells after n stages. A183061 (the first differences) gives the number of cells turned "ON" at the n-th stage.
%C A183060 Note that this is just half plus the rest of the center line of the cellular automaton described in A147562.
%C A183060 After 2^k stages the structure resembles an isosceles right triangle. For a three-dimensional version using cubes see A186410. For more information see A147562.
%H A183060 JungHwan Min, <a href="/A183060/b183060.txt">Table of n, a(n) for n = 0..2500</a>
%H A183060 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 A183060 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, pp. 31-32.
%H A183060 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 A183060 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A183060 a(n) = n + (A147562(n) - 1)/2, n >= 1.
%F A183060 a(n) = n + 2*A151920(n-2), n >= 2.
%F A183060 a(2^n) = A076024(n+1). - _Nathaniel Johnston_, Mar 14 2011
%e A183060 Illustration of the structure after eight stages in which we label the generations of cells turned ON by consecutive numbers:
%e A183060          8
%e A183060         878
%e A183060        8 6 8
%e A183060       8765678
%e A183060      8 8 4 8 8
%e A183060     878 434 878
%e A183060    8 6 4 2 4 6 8
%e A183060   876543212345678
%e A183060 ...................
%e A183060 There are 50 "ON" cells so a(8) = 50.
%t A183060 A183060[0] = 0; A183060[n_] := Total[With[{m = n - 1}, CellularAutomaton[{4042387958, 2, {{0, 1}, {-1, 0}, {0, 0}, {1, 0}, {0, -1}}}, {{{1}}, 0}, {{{m}}, -m}]], 2] (* _JungHwan Min_, Jan 24 2016 *)
%t A183060 A183060[0] = 0; A183060[n_] := Total[With[{m = n - 1}, CellularAutomaton[{686, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, {{{m}}, -m}]], 2] (* _JungHwan Min_, Jan 24 2016 *)
%Y A183060 Cf. A139250, A147562, A147582, A147610, A151920, A183061, A186410.
%K A183060 nonn
%O A183060 0,3
%A A183060 _Omar E. Pol_, Feb 20 2011
%E A183060 Comments edited by _Omar E. Pol_, Mar 19 2011 at the suggestion of _John W. Layman_ and _Franklin T. Adams-Watters_