A262620 -id:A262620 - cp's OEIS frontend

A262621 First differences of A262620.

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

Offset: 0

Omar E. Pol, Nov 03 2015

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

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

Cf. A147582, A160721, A261693, A262620.

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

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

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.)

Original entry on oeis.org

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

Omar E. Pol, Sep 25 2015

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.

			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.

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

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

cp's OEIS Frontend

A262621 First differences of A262620.

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