A173460 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

0, 1, 9, 21, 29, 81, 93, 105, 189, 225, 253, 441, 453, 465, 549, 585, 621, 873, 909, 945, 1197, 1305, 1397, 2025, 2037, 2049, 2133, 2169, 2205, 2457, 2493, 2529, 2781, 2889, 2997, 3753, 3789, 3825, 4077, 4185, 4293, 5049, 5157, 5265, 6021, 6345, 6637, 8649

Offset

0,3

Comments

On the infinite square grid, we start at stage 0 with all cells in OFF state. At stage 1, we turn ON a single cell, in the central position.

In order to construct this sequence we use the following rules:

- If n is congruent to 0 (mod 3), we turn "ON" the cells around the vertex of every convex corner formed in the structure at the generation n-1. Note that every vertex is surrounded by three new "ON" cells.

- If n is congruent to 1 (mod 3), we turn "ON" the possible bridge cells and the possible peninsula cells (For the definition of bridge cell and of peninsula cell see A160117).

- If n is congruent to 2 (mod 3), we turn "ON" the cells around the cells turned "ON" at the generation n-1.

- Everything that is already ON remains ON.

A173461, the first differences, gives the number of cells turned "ON" at n-th stage.

Links

Lars Blomberg, Table of n, a(n) for n = 0..6000

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.]

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Formula

a(0)=0, a(n) = a(n-1) + A173461(n), n>=1. - [Lars Blomberg, Apr 23 2013]

Example

Array begins:
0, 1, 9;
21, 29, 81;
93, 105, 189;
225, 253, 441;
453, 465, 549;
...
If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
7..7.....7..7
.66.......66.
.65555555556.
7.545545545.7
..553353355..
..553222355..
..545212545..
..553222355..
..553353355..
7.545545545.7
.65555555556.
.66.......66.
7..7.....7..7

Crossrefs

Cf. A160117, A160118, A173456, A173461, A173462, A173463.

Sequence in context: A039289 A045252 A139538 * A110701 A243703 A133929

Adjacent sequences: A173457 A173458 A173459 * A173461 A173462 A173463

Keyword

nonn

Author

Omar E. Pol, Feb 18 2010

Extensions

a(18)-a(47) from Lars Blomberg, Apr 23 2013

Status

approved