A261693 -id:A261693 - cp's OEIS frontend

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

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.

A262620 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton on the square grid (see Comments lines for definition).

Original entry on oeis.org

1, 5, 17, 21, 49, 69, 81, 85, 145, 197, 241, 277, 305, 325, 337, 341, 465, 581, 689, 789, 881, 965, 1041, 1109, 1169, 1221, 1265, 1301, 1329, 1349, 1361, 1365, 1617, 1861, 2097, 2325, 2545, 2757, 2961, 3157, 3345, 3525, 3697, 3861, 4017, 4165, 4305, 4437, 4561, 4677, 4785, 4885, 4977, 5061, 5137, 5205, 5265, 5317, 5361, 5397

Offset: 0

Omar E. Pol, Oct 16 2015

On the infinite square grid consider four 90-degree wedges forming a "X" with the vertex located at the center of a cell.
At stage 0 we start with an ON cell in the vertex of the wedges, so a(0) = 1.
In order to construct the structure we use the following rules for the South wedge:
- The cells turned ON remain ON forever.
- At stage 1 we turn ON the nearest cell to the initial ON cell.
- 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 wedge.
- 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 wedge, 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.
The structures in the other three wedges are copies of the structure in the South wedge but they grow in direction East, North and West.
Note that in every wedge the structure seems to grow into the holes of a virtual structure similar to the Sierpiński's triangle but using square cells.
A262621 gives the number of cells turned "ON" at n-th stage.
This is analog of A256266, but here we are working on the square grid and we have four wedges, not six wedges.

			Illustration of the structure after 15 generations:
.
.                                   O
.                                 O O O
.                               O O O O O
.                             O O O O O O O
.                           O O O O O O O O O
.                         O O O O O O O O O O O
.                       O O O O O O O O O O O O O
.                     O O O O O O O O O O O O O O O
.                   O               O               O
.                 O O             O O O             O O
.               O O O           O O O O O           O O O
.             O O O O         O O O O O O O         O O O O
.           O O O O O       O       O       O       O O O O O
.         O O O O O O     O O     O O O     O O     O O O O O O
.       O O O O O O O   O O O   O   O   O   O O O   O O O O O O O
.     O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
.       O O O O O O O   O O O   O   O   O   O O O   O O O O O O O
.         O O O O O O     O O     O O O     O O     O O O O O O
.           O O O O O       O       O       O       O O O O O
.             O O O O         O O O O O O O         O O O O
.               O O O           O O O O O           O O O
.                 O O             O O O             O O
.                   O               O               O
.                     O O O O O O O O O O O O O O O
.                       O O O O O O O O O O O O O
.                         O O O O O O O O O O O
.                           O O O O O O O O O
.                             O O O O O O O
.                               O O O O O
.                                 O O O
.                                   O
.
There are 341 ON cells in the structure, so a(15) = 341.
Note that every circle in the structure should be replaced with a square cell.

Cf. A147562, A160720, A256266, A261692, A261693, A262621.

a(n) = 1 + 4*A261692(n).

A262621 First differences of A262620.

Original entry on oeis.org

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.

A327923 Irregular triangle read by rows: Coefficients of Schick's polynomials P(n, y^2), for n >= 1.

Original entry on oeis.org

1, 1, -2, -1, 10, -24, 16, 1, -42, 504, -2640, 7040, -9984, 7168, -2048, -1, 170, -8568, 201552, -2687360, 22573824, -127339520, 502081536, -1417641984, 2901606400, -4310958080, 4600627200, -3435134976, 1702887424, -503316480, 67108864

Offset: 1

Wolfdieter Lang, Nov 19 2019

The length of row n of this irregular triangle t(n, k) is 2^(n-1) = A000079(n-1), n >= 1.
These polynomials P(n, y^2) = Sum_{k=0..2^(n-1)-1} t(n, k)*y^2 appear in table 2 (Tabelle 2), p. 157, of Carl Schick's book as x_n/(2^n*x*y), for n >= 1, and  x_0 = x. The polynomials y_n of Tabelle 1 on p. 156 appear as y_n = -Sum_{n=0..2^(n-1)} A321369(n, k)*y^(2*k), for n >= 1, with y_0 = y, and are related to the n-th iteration of the Chebyshev polynomial -T(2, y) = 1 - 2*x^2.
Originally the polynomials y_n and x_n are defined in Schick's rare notebook as: y_n = 1 - 2*(y_{n-1})^2, for n >= 1, with y_0 = y, and x_n/(2^n*x*y) = Product_{j=1..n-1} y_j, for n >= 1, and x_0 = x. The (x_n, y_n) come from coordinates of points on the unit circle with x_n = cos(phi_n) and y_n = sin(phi_n), with some initial condition (x_0, y_0) = (x, y).

			The irregular triangle T(n, k) begins:
n\k   0    1    2      3     4      5     6      7 ...
------------------------------------------------------
1:    1
2:    1   -2
3:   -1   10  -24     16
4:    1  -42  504  -2640  7040  -9984  7168  -2048
...
Row n = 5: -1  170  -8568  201552  -2687360  22573824  -127339520 502081536  -1417641984  2901606400  -4310958080 4600627200  -3435134976  1702887424  -503316480  67108864.
...
---------------------------------------------------------------------------

Carl Schick, Trigonometrie und unterhaltsame Zahlentheorie, Bokos Druck, Zürich, 2003 (ISBN 3-9522917-0-6).

Cf. A000079, A053120, A321369.

Row polynomials: P(n, y^2) = Product_{j=1..n-1} y_j(y^2), for n >= 1 (the empty product equals 1), where y_j(y^2) = -T^{[j]}(2, y) = -T(2^j, y), the j-th iteration of -T(2, y) = - 1 + 2*y^2, with Chebyshev's T polynomials (A053120).
Row polynomials linearized in T: P(n, y^2) = (-1)^(n-1)*(1/2^(n-2)) * Sum_{m=1..2^(n-2)} T(2*(2^(n-1) + 1 - 2*m), y), for n >= 2, and  P(1, y^2) = 1. See the irregular triangle A261693(n-1, m) = 2^(n-1) + 1 - 2*m, n >= 2, 1 <= m <= 2^(n-2).
Irregular triangle: t(n, k) = [y^(2*k)] P(n, y^2), n >= 1, k = 0, 1, ..., 2^(n-1)-1.

cp's OEIS Frontend

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.)

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A262620 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton on the square grid (see Comments lines for definition).

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A262621 First differences of A262620.

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A327923 Irregular triangle read by rows: Coefficients of Schick's polynomials P(n, y^2), for n >= 1.

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