cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A256266 Total number of ON states after n generations of cellular automaton based on triangles (see Comments lines for definition).

Original entry on oeis.org

0, 6, 18, 24, 48, 66, 78, 84, 132, 174, 210, 240, 264, 282, 294, 300, 396, 486, 570, 648, 720, 786, 846, 900, 948, 990, 1026, 1056, 1080, 1098, 1110, 1116, 1308, 1494, 1674, 1848, 2016, 2178, 2334, 2484, 2628, 2766, 2898, 3024, 3144, 3258, 3366, 3468, 3564, 3654, 3738, 3816, 3888, 3954, 4014, 4068, 4116, 4158, 4194, 4224, 4248
Offset: 0

Views

Author

Omar E. Pol, Mar 20 2015

Keywords

Comments

On the infinite triangular grid we start at stage 0 with a hexagon formed by six OFF cells, so a(0) = 0.
At stage 1, around the mentioned hexagon, six triangular cells connected by their vertices are turned ON forming a six-pointed star, so a(1) = 6.
We use the same rules as A255748 for every one of the six 60-degree wedges of the structure.
If n is a power of 2 minus 1 and n is greater than 2, then the structure looks like concentric six-pointed stars.
If n is a power of 2 and n is greater than 2, then the structure looks like a hexagon that contains concentric six-pointed stars.
Note that in every wedge the structure seems to grow into the holes of a virtual Sierpiński's triangle (see example).

Examples

			Illustration of the structure after 15 generations:
(Note that every circle should be replaced with a triangle.)
.
.                            O
.                           O O
.                          O O O
.                         O O O O
.                        O O O O O
.                       O O O O O O
.                      O O O O O O O
.                     O O O O O O O O
.    O O O O O O O O \       O       / O O O O O O O O
.     O O O O O O O   \     O O     /   O O O O O O O
.      O O O O O O     \   O O O   /     O O O O O O
.       O O O O O       \ O O O O /       O O O O O
.        O O O O O O O O \   O   / O O O O O O O O
.         O O O   O O O   \ O O /   O O O   O O O
.          O O     O O O O \ O / O O O O     O O
.           O       O   O O \ / O O   O       O
.            - - - - - - - -   - - - - - - - -
.           O       O   O O / \ O O   O       O
.          O O     O O O O / O \ O O O O     O O
.         O O O   O O O   / O O \   O O O   O O O
.        O O O O O O O O /   O   \ O O O O O O O O
.       O O O O O       / O O O O \       O O O O O
.      O O O O O O     /   O O O   \     O O O O O O
.     O O O O O O O   /     O O     \   O O O O O O O
.    O O O O O O O O /       O       \ O O O O O O O O
.                     O O O O O O O O
.                      O O O O 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 300 ON cells, so a(15) = 300.

Links

Crossrefs

Cf. A001316, A047999, A080079, A139250, A151723, A160120, A161330, A161644, A255748, A256256.

Programs

  • Mathematica
    6*Join[{0}, Accumulate@ Flatten@ Table[Range[2^n, 1, -1], {n, 0, 5}]] (* Michael De Vlieger, Nov 03 2022 *)

Formula

a(n) = 6 * A255748(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

Views

Author

Omar E. Pol, Sep 25 2015

Keywords

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

Crossrefs

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

Formula

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

A262617 First differences of A256266.

Original entry on oeis.org

0, 6, 12, 6, 24, 18, 12, 6, 48, 42, 36, 30, 24, 18, 12, 6, 96, 90, 84, 78, 72, 66, 60, 54, 48, 42, 36, 30, 24, 18, 12, 6, 192, 186, 180, 174, 168, 162, 156, 150, 144, 138, 132, 126, 120, 114, 108, 102, 96, 90, 84, 78, 72, 66, 60, 54, 48, 42, 36, 30, 24, 18, 12, 6, 384, 378, 372, 366, 360, 354, 348, 342, 336, 330, 324, 318
Offset: 0

Views

Author

Omar E. Pol, Oct 02 2015

Keywords

Comments

Number of cells turned ON at n-th stage of the cellular automaton of A256266.

Examples

			With the terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
0;
6;
12, 6;
24, 18, 12, 6;
48, 42, 36, 30, 24, 18, 12, 6;
96, 90, 84, 78, 72, 66, 60, 54, 48, 42, 36, 30, 24, 18, 12, 6;
...
Apart from the initial zero the rows list the initial terms of the positive multiples of 6 in decreasing order.

Links

Crossrefs

Cf. A008588, A011782, A080079, A255748, A256257, A256266.

Programs

  • PARI
    a(n) = if(n==0, 0, -6*n+12*2^floor(log(n)/log(2)));
vector(100, n, a(n-1)) \\ Altug Alkan, Oct 04 2015

Formula

a(n) = 6 * A080079(n), n >= 1.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.