login
A173460
Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).
5
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
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.]
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
KEYWORD
nonn
AUTHOR
Omar E. Pol, Feb 18 2010
EXTENSIONS
a(18)-a(47) from Lars Blomberg, Apr 23 2013
STATUS
approved