A262620 -id:A262620

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A262621

First differences of A262620.

+20
2

1, 4, 12, 4, 28, 20, 12, 4, 60, 52, 44, 36, 28, 20, 12, 4, 124, 116, 108, 100, 92, 84, 76, 68, 60, 52, 44, 36, 28, 20, 12, 4, 252, 244, 236, 228, 220, 212, 204, 196, 188, 180, 172, 164, 156, 148, 140, 132, 124, 116, 108, 100, 92, 84, 76, 68, 60, 52, 44, 36, 28, 20, 12, 4, 508, 500, 492, 484, 476, 468, 460, 452, 444, 436

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Number of cells turned "ON" at n-th stage of cellular automaton of A262620.

LINKS

Table of n, a(n) for n=0..73.

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) = 4 * A261693(n), n >= 1.

EXAMPLE

With the terms written as an irregular triangle in which row lengths are the terms of A011782 the sequence begins:

12, 4;

28, 20, 12, 4;

60, 52, 44, 36, 28, 20, 12, 4;

124, 116, 108, 100, 92, 84, 76, 68, 60, 52, 44, 36, 28, 20, 12, 4;

...

CROSSREFS

Cf. A147582, A160721, A261693, A262620.

Row sums give A000302. Row lengths give A011782. Right border gives A123932. Column 1 is A173033.

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Nov 03 2015

STATUS

approved

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.)

+10
4

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.

KEYWORD

nonn,look

AUTHOR

Omar E. Pol, Sep 25 2015

STATUS

approved

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