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 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.
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
FORMULA
a(0)=0, a(n)=a(n-1)+A173457(n), n>=1
EXAMPLE
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Omar E. Pol, Feb 18 2010
EXTENSIONS
a(41)-a(48) from Lars Blomberg, Apr 22 2013
STATUS
approved