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

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

Offset: 0

Author

Omar E. Pol, Feb 20 2011

Keywords

• nonn

Comments

On the semi-infinite 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 n-th stage.

Note that this is just half plus the rest of the center line of the cellular automaton described in A147562.

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.

Examples

			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.
		

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), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, pp. 31-32.
• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to cellular automata

Crossrefs

Cf. A139250, A147562, A147582, A147610, A151920, A183061, A186410.

Programs

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 *)

Formula

a(n) = n + (A147562(n) - 1)/2, n >= 1.

a(n) = n + 2*A151920(n-2), n >= 2.

a(2^n) = A076024(n+1). - Nathaniel Johnston, Mar 14 2011

Extensions

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