A261692 Number of "ON" cells after n-th stage in a cellular automaton in a 90-degree wedge on the square grid. (See Comments lines for definition.)

0, 1, 4, 5, 12, 17, 20, 21, 36, 49, 60, 69, 76, 81, 84, 85, 116, 145, 172, 197, 220, 241, 260, 277, 292, 305, 316, 325, 332, 337, 340, 341, 404, 465, 524, 581, 636, 689, 740, 789, 836, 881, 924, 965, 1004, 1041, 1076, 1109, 1140, 1169, 1196, 1221, 1244, 1265, 1284, 1301, 1316, 1329, 1340, 1349, 1356, 1361, 1364, 1365, 1492

Offset: 0

Author

Omar E. Pol, Sep 25 2015

Keywords

• nonn
• look

Comments

In order to construct the structure we use the following rules:

- On the square grid we are in a 90-degree wedge with the vertex located on top of the wedge.

- At stage 0 there are no ON cells, so a(0) = 0.

- At stage 1 we turn ON the nearest cell of the vertex, so a(1) = 1.

- The cells turned ON remain ON forever.

- If n is a power of 2, at stage n we turn "ON" 2*n - 1 connected cells in the n-th row of the structure.

- Otherwise, if n is not a power of 2, at stage n we turn "ON" k - 2 connected cells in the n-th row of the structure, where k is the number of ON cells in row n - 1.

- The "ON" cells of row n must be centered respect to the "ON" cells of row n - 1.

Note that the structure seems to grow into the holes of a virtual structure similar to the Sierpiński's triangle but using square cells (see example).

A261693 gives the number of cells turned "ON" at n-th stage.

This is analog of A255748, but here we are working on the square grid.

Examples

			Illustration of initial terms (n = 0..15):
------------------------------------------------------
n  A261692(n)  a(n)                Diagram
------------------------------------------------------
0      0        0                    /_\
1      1        1                  /_|_|_\
2      3        4                / |_|_|_| \
3      1        5              /_ _ _|_|_ _ _\
4      7       12            / |_|_|_|_|_|_|_| \
5      5       17          /     |_|_|_|_|_|     \
6      3       20        /         |_|_|_|         \
7      1       21      /_ _ _ _ _ _ _|_|_ _ _ _ _ _ _\
8     15       36    / |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_| \
9     13       49        |_|_|_|_|_|_|_|_|_|_|_|_|_|
10    11       60          |_|_|_|_|_|_|_|_|_|_|_|
11     9       69            |_|_|_|_|_|_|_|_|_|
12     7       76              |_|_|_|_|_|_|_|
13     5       81                |_|_|_|_|_|
14     3       84                  |_|_|_|
15     1       85                    |_|
...
After 15 generations there are 85 ON cells in the structure, so a(15) = 85.
		

Links

• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to cellular automata

Crossrefs

Cf. A001316, A047999, A139250, A147562, A255748, A261693, A262620.

Formula

a(n) = (A262620(n) - 1)/4.