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.

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A255748 Total number of ON states after n generations of cellular automaton based on triangles in a 60-degree wedge (see Comments lines for definition).

Original entry on oeis.org

1, 3, 4, 8, 11, 13, 14, 22, 29, 35, 40, 44, 47, 49, 50, 66, 81, 95, 108, 120, 131, 141, 150, 158, 165, 171, 176, 180, 183, 185, 186, 218, 249, 279, 308, 336, 363, 389, 414, 438, 461, 483, 504, 524, 543, 561, 578, 594, 609, 623, 636, 648, 659, 669, 678, 686, 693, 699, 704, 708, 711, 713, 714, 778, 841, 903, 964, 1024
Offset: 1

Views

Author

Omar E. Pol, Mar 30 2015

Keywords

Comments

Also partial sums of A080079.
In order to construct the structure we use the following rules:
On the infinite triangular grid we are in a 60-degree wedge with the vertex located on top of the wedge.
The nearest triangular cell to the vertex remains OFF.
At stage 1, we turn ON the cell whose base is adjacent to the previous OFF cell.
At stage n, in the n-th level of the structure, we turn ON k cells connected by their vertices with their bases up, where k = A080079(n).
The cells turned ON remain ON forever.
The structure seems to grow into the holes of a virtual Sierpiński's triangle (see example).
Note that this is also the structure in every one of the six wedges of the structure of A256266.
A080079 gives the number of cells turned ON at n-th stage.

Examples

			Illustration of initial terms:
-----------------------------------------------------------
n   A080079(n)   a(n)                  Diagram
-----------------------------------------------------------
.                                        / \
1       1         1                     / T \
2       2         3                    / T T \
3       1         4                   /   T   \
4       4         8                  / T T T T \
5       3        11                 /   T T T   \
6       2        13                /     T T     \
7       1        14               /       T       \
8       8        22              / T T T T T T T T \
9       7        29             /   T T T T T T T   \
10      6        35            /     T T T T T T     \
11      5        40           /       T T T T T       \
12      4        44          /         T T T T         \
13      3        47         /           T T T           \
14      2        49        /             T T             \
15      1        50       /               T               \
...
For n = 15 after 15 generations there are 50 ON cells in the structure, so a(15) = 50.

Links

Crossrefs

Cf. A047999, A001316, A080079, A139250, A169779, A169788, A170905, A233970, A256256, A256266.

Programs

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

Formula

a(n) = A256266(n)/6.

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.

A256257 6 times numbers of Gould's sequence A001316.

Original entry on oeis.org

6, 12, 12, 24, 12, 24, 24, 48, 12, 24, 24, 48, 24, 48, 48, 96, 12, 24, 24, 48, 24, 48, 48, 96, 24, 48, 48, 96, 48, 96, 96, 192, 12, 24, 24, 48, 24, 48, 48, 96, 24, 48, 48, 96, 48, 96, 96, 192, 24, 48, 48, 96, 48, 96, 96, 192, 48, 96, 96, 192, 96, 192, 192, 384, 12, 24, 24, 48, 24, 48, 48, 96, 24, 48, 48, 96, 48, 96, 96, 192
Offset: 0

Views

Author

Omar E. Pol, Mar 20 2015

Keywords

Comments

Also, number of triangular cells turned ON at (n+1)-st stage in the structure of A256256.
First differences of A256256.

Examples

			Written as an irregular triangle in which the row lengths are the terms of A011782, the sequence begins:
6;
12;
12, 24;
12, 24, 24, 48;
12, 24, 24, 48, 24, 48, 48, 96;
12, 24, 24, 48, 24, 48, 48, 96, 24, 48, 48, 96, 48, 96, 96, 192;
...

Links

Crossrefs

Cf. A001316, A011782, A047999, A048896, A117973, A151787, A160713, A160721, A256256.

Formula

a(n) = 6*A001316(n) = 3*A117973(n) = 2*A160713(n).
a(n) = 12*A048896(n-1), n >= 1.
Showing 1-3 of 3 results.

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