

A183060


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


4



0, 1, 4, 7, 14, 17, 24, 31, 50, 53, 60, 67, 86, 93, 112, 131, 186, 189, 196, 203, 222, 229, 248, 267, 322, 329, 348, 367, 422, 441, 496, 551, 714, 717, 724, 731, 750, 757, 776, 795, 850, 857, 876, 895, 950, 969, 1024, 1079, 1242, 1249, 1268, 1287
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OFFSET

0,3


COMMENTS

On the semiinfinite square grid, start with all cells OFF.
Turn a single cell to the ON state in row 1.
At each subsequent step, each cell with exactly one neighbor ON is turned ON, and everything that is already ON remains ON.
The sequence gives the number of "ON" cells after n stages. A183061 (the first differences) gives the number of cells turned "ON" at the nth stage.
Note that this is just half plus the rest of the center line of the cellular automaton described in A147652.
After 2^k stages the structure resembles an isosceles right triangle. For a threedimensional version using cubes see A186410. For more information see A147562.


LINKS

JungHwan Min, Table of n, a(n) for n = 0..2500
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n1)1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata


FORMULA

a(n) = n + (A147562(n)  1)/2, n >= 1.
a(n) = n + 2*A151920(n2), n >= 2.
a(2^n) = A076024(n+1).  Nathaniel Johnston, Mar 14 2011


EXAMPLE

Illustration of the structure after eight stages in which we label the generations of cells turned ON by consecutive numbers:
8
878
8 6 8
8765678
8 8 4 8 8
878 434 878
8 6 4 2 4 6 8
876543212345678
...................
There are 50 "ON" cells so a(8) = 50.


MATHEMATICA

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 *)
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 *)


CROSSREFS

Cf. A139250, A147562, A147582, A147610, A151920, A183061, A186410.
Sequence in context: A310907 A310908 A159912 * A310909 A310910 A310911
Adjacent sequences: A183057 A183058 A183059 * A183061 A183062 A183063


KEYWORD

nonn


AUTHOR

Omar E. Pol, Feb 20 2011


EXTENSIONS

Comments edited by Omar E. Pol, Mar 19 2011 at the suggestion of John W. Layman and Franklin T. AdamsWatters


STATUS

approved



