A173456 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

0, 1, 9, 21, 25, 53, 89, 93, 121, 157, 169, 253, 361, 365, 393, 429, 441, 525, 633, 645, 729, 837, 873, 1125, 1449, 1453, 1481, 1517, 1529, 1613, 1721, 1733, 1817, 1925, 1961, 2213, 2537, 2549, 2633, 2741, 2777, 3029, 3353, 3389, 3641, 3965, 4073, 4829, 5801

Offset: 0

Author

Omar E. Pol, Feb 18 2010

Keywords

• nonn

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 peninsula cells (For the definition 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.

A173457, the first differences, gives the number of cells turned "ON" at n-th stage.

Examples

			Array begins:
0, 1, 9;
21, 25, 53;
89, 93, 121;
157, 169, 253;
361, 365, 393;
...
If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
7...........7
.66.66.66.66.
.65556.65556.
..545...545..
.65533.33556.
.66.32223.66.
.....212.....
.66.32223.66.
.65533.33556.
..545...545..
.65556.65556.
.66.66.66.66.
7...........7
		

Links

• Lars Blomberg, Table of n, a(n) for n = 0..5999
• 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

Crossrefs

Cf. A160117, A160118, A173457, A173458, A173459, A173460.

Formula

a(0)=0, a(n)=a(n-1)+A173457(n), n>=1

Extensions

a(41)-a(48) from Lars Blomberg, Apr 22 2013