A151723 -id:A151723 - cp's OEIS frontend

A222180 Total number of ON states after n generations of cellular automaton based on pentagons. Also P-toothpick sequence (see Comments lines for definition).

0, 1, 6, 16, 26, 36, 56, 86, 106, 116, 136, 176, 216, 246, 296, 366, 406, 416, 436, 476, 536, 616

Offset: 0

Omar E. Pol, Mar 15 2013

Analog of A161644, A147562 and A151723, but here we are working without a lattice. Each regular pentagon has five virtual neighbors. Overlapping are prohibited. The sequence gives the number of pentagons in the structure after n-th stage. A222181 (the first differences) gives the number of pentagons added at n-th stage.

Also this is a P-toothpick sequence since every pentagon can be replaced by a P-toothpick which is formed by five toothpicks as a five-pointed star. Note that each toothpick can be represented as an apothem or as a radius of a pentagon. In both types of structures the number of toothpicks after n-th stage is equal to 5*a(n).

Omar E. Pol, Illustration of structure after 5 stages, (Contains 36 pentagons).
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A147562, A151723, A152968, A161330, A161644, A182632, A182840, A222172, A222176, A222181.

a(n) = 6 + 10*A222172(n-2), n >= 2. - Omar E. Pol, Nov 24 2013

Name improved by Omar E. Pol, Nov 24 2013

A342273 Consider the k-th row of triangle A170899 starting at the 3 in the middle of the row; the row from that point on converges to this sequence as k increases.

Original entry on oeis.org

3, 6, 11, 13, 13, 21, 33, 29, 17, 21, 37, 51, 51, 57, 77, 61, 25, 21, 37, 51, 55, 71, 111, 127, 91, 65, 93, 137, 143, 147, 175, 127, 41, 21, 37, 51, 55, 71, 111, 127, 95, 79, 119, 179, 207, 219, 271, 279, 171, 81, 93, 137, 159, 195, 291, 363

Offset: 0

N. J. A. Sloane, Mar 13 2021

nonn

It would be nice to have a formula or recurrence for any of A170899, A342272-A342278, or any nontrivial relation between them. This might help to understand the fractal structure of the mysterious hexagonal Ulam-Warburton cellular automaton A151723.

Needs a bigger b-file.

			Row k=6 of A170899 breaks up naturally into 7 pieces:
1;
2;
4,4;
4,8,12,8;
4,8,14,18,16,20,28,16;
4,8,14,18,18,26,42,42,24,20,36,50,46,50,62,32;
3,6,11,13,13,21,33,29,17,21,37,51,51,57,77,61,21,15,27,34,36,52,80,80,44,38,62,81,58,73,63,0.
The last piece already matches the sequence for 16 terms. The number of matching terms doubles at each row.

N. J. A. Sloane, Table of n, a(n) for n = 0..255

Cf. A151723, A151724, A342272.

A170906 Triangle read by rows: T(n,k) = number of cells that are turned from OFF to ON at stage k of the cellular automaton in the 30-60-90 triangle of hexagons defined in Comments.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 4, 1, 2, 1, 1, 2, 2, 4, 2, 2, 3, 3, 1, 1, 2, 2, 4, 2, 4, 5, 4, 1, 2, 1, 1, 2, 2, 4, 2, 4, 6, 6, 1, 2, 3, 3, 1, 1, 2, 2, 4, 2, 4, 6, 8, 1, 2, 3, 5, 3, 3, 1, 1, 2, 2, 4, 2, 4, 6, 8, 2, 2, 3, 5, 5, 3, 5, 4, 1, 1, 2, 2, 4, 2, 4, 6, 8, 2, 4, 5, 6, 7, 6, 6, 4, 1, 2, 1

Offset: 1

N. J. A. Sloane, Jan 24 2010

Consider the tiling of the plane by hexagons, where each cell has 6 neighbors, as in the A151723, A151724, A170905.
Assume the hexagons are oriented so that each one has a pair of vertical edges.
Consider the (30 deg., 60 deg., 90 deg.) triangle of hexagons with n hexagons along the short side, along the X-axis, 2n-1 hexagons along the hypotenuse and n hexagons separated by single edges along the middle side, along the Y-axis.
Initially all cells are OFF. At stage 1, the cell in the 60-degree corner is turned ON; thereafter, a cell is turned ON if it has exactly one ON neighbor in the triangle. Once a cell is ON it stays ON.
T(n,k) is the number of cells that are turned from OFF to ON at stage k (1 <= k <= 2n-1).
The rows converge to A170905. The rows sums give A170907.
Row n contains 2n-1 terms.
I wish I had a recurrence for this sequence!

			Triangle begins:
1
1 2 1
1 2 2 2 1
1 2 2 4 1 2 1
1 2 2 4 2 2 3 3 1
1 2 2 4 2 4 5 4 1 2 1
1 2 2 4 2 4 6 6 1 2 3 3 1
1 2 2 4 2 4 6 8 1 2 3 5 3 3 1
1 2 2 4 2 4 6 8 2 2 3 5 5 3 5 4 1
1 2 2 4 2 4 6 8 2 4 5 6 7 6 6 4 1 2 1
...
Row n = 4, [1 2 2 4 1 2 1], corresponds to the sequence of cells being turned ON shown in the following triangle (X denotes a cell that stays OFF). The hexagons have to be imagined.
7
.6
6.5
.X.4
X.4.3
.4.X.2
4.3.2.1

N. J. A. Sloane, Table of n, a(n) for n = 1..16384 (Rows 1..128, flattened)
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A151723, A151724, A170905, A170907, A169782 (partial sums across rows).

A253770 Number of ON states after n generations of cellular automaton based on triangles, with diamonds.

Original entry on oeis.org

0, 6, 24, 42, 96, 114, 168, 222, 348, 402, 456, 510, 636, 726, 852, 1014, 1320, 1482, 1536, 1590, 1716, 1806, 1932, 2094, 2400, 2598, 2724, 2886, 3192, 3498, 3840, 4254, 4956, 5442, 5568, 5622, 5748, 5838, 5964, 6126, 6432, 6630, 6756, 6918, 7224, 7530, 7872, 8286

Offset: 0

Omar E. Pol, Jan 11 2015

nonn

Also 6 times the Y-toothpicks sequence A160120.
Explanation: consider the Y-toothpick structure of A160120, then replace every Y-toothpick with six ON cells forming a star with three rhombuses (or diamonds) that share only one vertex. Every diamond contains two triangular cells that share one edge.
The rules are the essentially the same as A160120.
An ON cell remains ON forever.
The sequence gives the number of triangular ON cells after the n-th stage.
A253771 (the first differences) give the number of triangular cells turned "ON" at the n-th stage.
A160120 (the Y-toothpick sequence) gives the number of stars in the structure after the n-th stage.
A160121 gives the number of stars added at the n-th stage.
A160167 gives the number of diamonds in the structure after the n-th stage.

			After one generation, the cellular automaton looks like a star or a flower with three petals as shown below:
.
.        /\
.       _\/_
.      /_/\_\
.
There are one star, three diamonds and six ON cells, so a(1) = 6.

Cf. A139250, A147562, A151723, A160120, A160157, A160167, A161644, A182632, A250300, A253771.

a(n) = 6*A160120(n) = 3*A160157(n) = 2*A160167(n).

A256256 Total number of ON cells after n generations of cellular automaton on triangular grid, starting from a node, in which every 60-degree wedge looks like the Sierpiński's triangle.

Original entry on oeis.org

0, 6, 18, 30, 54, 66, 90, 114, 162, 174, 198, 222, 270, 294, 342, 390, 486, 498, 522, 546, 594, 618, 666, 714, 810, 834, 882, 930, 1026, 1074, 1170, 1266, 1458, 1470, 1494, 1518, 1566, 1590, 1638, 1686, 1782, 1806, 1854, 1902, 1998, 2046, 2142, 2238, 2430, 2454, 2502, 2550, 2646, 2694, 2790, 2886, 3078, 3126, 3222, 3318, 3510, 3606, 3798, 3990, 4374

Offset: 0

Omar E. Pol, Mar 20 2015

nonn

Analog of A160720, but here we are working on the triangular lattice.
The first differences (A256257) gives the number of triangular cells turned ON at every generation.
Also 6 times the sum of all entries in rows 0 to n of Sierpiński's triangle A047999.
Also 6 times the total number of odd entries in first n rows of Pascal's triangle A007318, see formula section.
The structure contains three distinct kinds of polygons formed by triangular ON cells: the initial hexagon, rhombuses (each one formed by two ON cells) and unit triangles.
Note that if n is a power of 2 greater than 2, the structure looks like concentric hexagons with triangular holes, where some of them form concentric stars.

			On the infinite triangular grid we start with all triangular cells turned OFF, so a(0) = 0.
At stage 1, in the structure there are six triangular cells turned ON forming a regular hexagon, so a(1) = 6.
At stage 2, there are 12 new triangular ON cells forming six rhombuses around the initial hexagon, so a(2) = 6 + 12 = 18.
And so on.

Michael De Vlieger, Table of n, a(n) for n = 0..16386
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 30.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A001316, A006046, A007318, A025192, A047999, A151723, A160120, A160720, A161644, A256257.

Prepend[6*FoldList[Plus, 0, Total /@ CellularAutomaton[90, Join[Table[0, {#}], {1}, Table[0, {#}]], #]][[2 ;; -1]], 0] &[63] (* Michael De Vlieger, Nov 03 2022, after Bradley Klee at A006046 *)

a(n) = 6*sum(j=0, n, sum(k=0, j, binomial(j, k) % 2)); \\ Michel Marcus, Apr 01 2015

a(n) = 6*A006046(n).

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

Omar E. Pol, Mar 20 2015

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).

			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.

Michael De Vlieger, Table of n, a(n) for n = 0..16384
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 37.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

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

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

a(n) = 6 * A255748(n), n >= 1.

A169787 The rows of the triangle in A169786 converge to this sequence.

Original entry on oeis.org

1, 2, 3, 5, 5, 5, 9, 13, 9, 5, 9, 15, 19, 17, 21, 29, 17, 5, 9, 15, 19, 19, 27, 43, 43, 25, 21, 37, 51, 47, 51, 63, 33, 5, 9, 15, 19, 19, 27, 43, 43, 27, 27, 47, 67, 71, 75, 99, 91, 41, 21, 37, 51, 55, 71, 111, 127, 87, 59, 87, 125, 119, 119, 133, 65, 5, 9, 15, 19, 19

Offset: 1

David Applegate and N. J. A. Sloane, May 12 2010

nonn

Cf. A169786, A151723, A342272 (another version: subtract 1 from each term).

A169788 a(n) = n(n+1)/2 - A169780(n).

Original entry on oeis.org

0, 0, 0, 1, 1, 4, 6, 7, 7, 14, 20, 25, 27, 30, 36, 37, 37, 52, 66, 79, 89, 100, 112, 117, 115, 122, 140, 153, 157, 158, 168, 167, 167, 198, 228, 257, 283, 310, 338, 359, 373, 396, 428, 453, 467, 474, 486, 491, 481, 496, 538, 575, 603, 628, 654, 665, 653, 650, 680, 707, 711

Offset: 0

N. J. A. Sloane, May 13 2010

nonn

If n is a power of 2, this is the number of OFF cells after n stages in a 60-degree wedge of the hexagonal CA (see A170905, A169780, A151723, A169789).

N. J. A. Sloane, Table of n, a(n) for n = 0..1025

Cf. A170905, A169780, A151723, A169789.

A250300 Number of ON states after n generations of cellular automaton based on triangles (compare A161644).

Original entry on oeis.org

0, 3, 6, 12, 24, 36, 42, 54, 72, 90, 108, 126, 162, 198, 210, 234, 264, 282, 300, 324, 366, 420, 462, 498, 558, 624, 678, 726, 816, 906, 936, 990, 1044, 1062, 1080, 1104, 1146, 1200, 1242, 1284, 1350, 1428, 1506, 1584, 1698, 1848, 1950, 2022, 2130

Offset: 0

Omar E. Pol, Jan 15 2015

nonn

The same rules as A161644 but here we start with three ON cells which share only one vertex.

Cf. A139250, A147562, A151723, A160120, A161644, A182632, A253770.

A335794 Number of ON cells at n-th generation in an "Ulam-Warburton and Friedkin Replicator" hybrid two-dimensional cellular automaton.

Original entry on oeis.org

0, 1, 7, 13, 31, 37, 67, 85, 139, 145, 175, 205, 283, 301, 391, 445, 595, 601, 631, 661, 739, 769, 895, 973, 1195, 1213, 1303, 1393, 1627, 1681, 1927, 2077, 2467, 2473, 2503, 2533, 2611, 2641, 2767, 2845, 3067, 3097, 3223, 3349, 3667, 3745, 4111, 4333, 4939, 4957

Offset: 0

Cody B Duncan, Jun 23 2020

This 2D CA uses the neighborhood:
[0 X X]
[X X X]
[X X 0]
If a cell has an even number of ON neighbors and it is currently OFF, stay OFF; otherwise turn ON.
The results are similar to those for A151723, but with a distorted grid.
(The "look" keyword refers to the animation. - N. J. A. Sloane, Jul 03 2020)

S. M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 (see Example 6, page 224).

Cody B. Duncan, Animation

Cf. A102900, A147562, A151723, A247666, A335795.

cp's OEIS Frontend

A222180 Total number of ON states after n generations of cellular automaton based on pentagons. Also P-toothpick sequence (see Comments lines for definition).

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A342273 Consider the k-th row of triangle A170899 starting at the 3 in the middle of the row; the row from that point on converges to this sequence as k increases.

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A170906 Triangle read by rows: T(n,k) = number of cells that are turned from OFF to ON at stage k of the cellular automaton in the 30-60-90 triangle of hexagons defined in Comments.

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A253770 Number of ON states after n generations of cellular automaton based on triangles, with diamonds.

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A256256 Total number of ON cells after n generations of cellular automaton on triangular grid, starting from a node, in which every 60-degree wedge looks like the Sierpiński's triangle.

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

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A169787 The rows of the triangle in A169786 converge to this sequence.

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A169788 a(n) = n(n+1)/2 - A169780(n).

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A250300 Number of ON states after n generations of cellular automaton based on triangles (compare A161644).

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A335794 Number of ON cells at n-th generation in an "Ulam-Warburton and Friedkin Replicator" hybrid two-dimensional cellular automaton.

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