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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.) 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

LINKS

Table of n, a(n) for n=0..64.

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

Index entries for sequences related to cellular automata

FORMULA

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

EXAMPLE

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.

CROSSREFS

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

Sequence in context: A269227 A103650 A131116 * A131328 A054451 A309479

Adjacent sequences:  A261689 A261690 A261691 * A261693 A261694 A261695

KEYWORD

nonn,look

AUTHOR

Omar E. Pol, Sep 25 2015

STATUS

approved

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Last modified November 16 17:25 EST 2019. Contains 329201 sequences. (Running on oeis4.)