A151723 -id:A151723 - cp's OEIS frontend

A334228 a(n) is the n-th completed odd-numbered level 2k-1 containing k ON-cells in a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^j >= 2k-1 generations of the automaton in A151723 have been computed).

1, 3, 7, 11, 15, 19, 23, 27, 31, 35, 43, 47, 51, 55, 59, 63, 67, 83, 91, 95, 99, 107, 111, 115, 119, 123, 127, 131, 163, 179, 187, 191, 195, 211, 219, 223, 227, 235, 239, 243, 247, 251, 255, 259, 323, 355, 371, 379, 383, 387, 419, 435, 443, 447, 451, 467, 475, 479, 483, 491, 495, 499, 503, 507

Offset: 1

Hartmut F. W. Hoft, Apr 19 2020

nonn

			The sequence appears to be the triangle L(m, j) below read by rows where each row m >= 2 contains the level numbers in the sequence between 2^m - 1 and 2^(m+1) - 5:
m/j  0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
1:   1
2:   3
3:   7  11
4:  15  19  23  27
5:  31  35  43  47  51  55  59
6:  63  67  83  91  95  99 107 111 115 119 123
7: 127 131 163 179 187 191 195 211 219 223 227 235 239 243 247 251
...
If T(i) = (i-2)*(i-1)/2 then row m >= 2 contains T(m) + 1 values.
The difference structure of this triangle L(m, j) is given by the triangle P(m, 1) = 4 and P(m, j) = 2^(m+2-j), for 2 <= j <= m, of powers of 2 as follows:
m/j  1  2  3  4  5
1:   4
2:   4  4
3:   4  8  4
4:   4 16  8  4
5:   4 32 16  8  4
...
Applying function log_2(k) - 1 to an entry k in this triangle gives the corresponding entry in the triangle of A193592.
Going backwards in triangle P(m, j) from the row labeled m - 2, left to right up to its vertex, starting with 2^m - 1 and computing the cumulative differences using the entries in triangle P(m, j) produces the numbers in row m of triangle L(m, j).

Hartmut F. W. Hoft, Plot of 3 rows of triangle L

Cf. A002024, A002260, A004736, A151723, A193592.

row[1] = 1; row[2] = 3; row[n_] := (2^n - 1) + Prepend[Accumulate[Flatten[Table[If[i==0||==j, 4, 2^(2+j-i)], {j, n-3, 0, -1}, {i, 0, j}]]], 0]/;n>=3
a334228[n_] := Flatten[Map[row,Range[n]]] (* first n rows in triangle L *)
a334228[8] (* sequence data *)

L(m, 0) = 2^(m-1); L(m, j) = 2^(m-1) + Sum_{i=k+1-j..k} P( t_1(i), t_2(i) ), for m >= 3 and for j=1..k, where k = (m-2)*(m-1)/2. Functions t_1(n) = floor(1/2 + sqrt(2*n)), A002024, and t_2(n) = binomial(floor(3/2 + sqrt(2*n)), 2) - n + 1, A004736, by Michael Somos, Jul 12 2003, are listed in triangle #7 in his link in A002260.

The formula for the count of ON-cells was verified through level 16384.

A182840 Toothpick sequence on hexagonal net.

Original entry on oeis.org

0, 1, 5, 13, 27, 43, 57, 81, 119, 151, 165, 189, 235, 299, 353, 409, 495, 559, 573, 597, 643, 707, 769, 849, 975, 1119, 1205, 1261, 1371, 1539, 1697, 1841, 2039, 2167, 2181, 2205, 2251, 2315, 2377, 2457, 2583, 2727, 2821, 2901, 3043, 3267, 3505, 3729, 4015

Offset: 0

Omar E. Pol, Dec 09 2010

nonn

Rules:
- Each new toothpick must lie on the hexagonal net such that the toothpick endpoints coincide with two consecutive nodes.
- Each exposed endpoint of the toothpicks of the old generation must be touched by the endpoints of two toothpicks of new generation.
The sequence gives the number of toothpicks after n stages. A182841 (the first differences) gives the number added at the n-th stage.
The toothpick structure has polygons in which there are uncovered grid points, the same as A160120 and A161206. For more information see A139250.
Has a behavior similar to A151723, A182632. - Omar E. Pol, Feb 28 2013
From Omar E. Pol, Feb 17 2023: (Start)
Assume that every triangular cell has area 1.
It appears that the structure contains only three types of polygons:
- Regular hexagons of area 6.
- Concave decagons (or concave 10-gons) of area 12.
- Concave dodecagons (or concave 12-gons) of area 18.
There are infinitely many of these polygons.
The structure contains concentric hexagonal rings formed by hexagons and also contains concentric hexagonal rings formed by alternating decagons and dodecagons.
For an animation see the movie version in the Links section.
The animation shows the fractal-like behavior the same as in other members of the family of toothpick cellular automata.
The structure has internal growth.
For another version starting from a node see A182632.
For a version of the structure in the first quadrant but on the square grid see A182838. (End)

			We start at stage 0 with no toothpicks.
At stage 1 we place a toothpick anywhere in the plane (For example, in vertical position). There are two exposed endpoints, so a(1)=1.
At stage 2 we place 4 toothpicks. Two new toothpicks touching each exposed endpoint. So a(2)=1+4=5. There are 4 exposed endpoints.
At stage 3 we place 8 toothpicks. a(3)=5+8=13. The structure has 8 exposed endpoints.
At stage 4 we place 14 toothpicks (Not 16) because there are 4 endpoints that are touched by new 8 toothpicks but there are 4 endpoints that are touched by only 6 new toothpicks (not 8), so a(4)=13+14=27.
After 4 stages the toothpick structure has 4 hexagons and 8 exposed endpoints.

Olaf Voß, Table of n, a(n) for n = 0..1000
David Applegate, The movie version
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
Olaf Voß, Illustration of initial terms
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

Cf. A139250, A160120, A161206, A182632, A182634, A182838.

More terms from Olaf Voß, Dec 24 2010

Wiki link added by Olaf Voß, Jan 14 2011

A151725 Number of ON states after n generations of cellular automaton rule described by the rulestring B1/S012345678.

Original entry on oeis.org

0, 1, 9, 13, 33, 37, 57, 77, 121, 125, 145, 165, 209, 237, 297, 373, 465, 469, 489, 509, 553, 581, 641, 717, 809, 837, 897, 981, 1097, 1213, 1409, 1645, 1833, 1837, 1857, 1877, 1921, 1949, 2009, 2085, 2177, 2205, 2265, 2349, 2465, 2581, 2777, 3013

Offset: 0

David Applegate and N. J. A. Sloane, Jun 13 2009

nonn

A cell is turned ON if exactly one of its eight neighbors is ON. An ON cell remains ON forever.
We start with a single ON cell.
Analog of A147562, which is the case when each cell has only four neighbors.
The equivalent Mathematica cellular automaton is obtained with neighborhood weights {{1,1,1},{1,9,1},{1,1,1}}, rule number 261634, and starting configuration {{1}}. [John W. Layman, Sep 11 2009]
Observation: Visual pattern similar to the toothpick structure (see A139250). [Omar E. Pol, Dec 14 2009]

David Applegate, Table of n, a(n) for n = 0..1000
David Applegate, The movie version
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. H. Packard and S. Wolfram, Two-Dimensional Cellular Automata, Journal of Statistical Physics, 38 (1985), 901-946. (See page 920, Figure 7e). Alternative copy
Rémy Sigrist, Illustration of the structure at stage 513
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A139250, A151726, A147562, A147582, A151723, A151735, A151747, A151728.

See A151731, A151732, A151733, A151734 for the same CA except that two neighbors must be ON for a cell to turn ON.

RasterGraphics[state_?MatrixQ, colors_Integer : 2, opts___] := Graphics[Raster[ Reverse[1 - state/(colors - 1)]], AspectRatio -> (AspectRatio /. {opts} /. AspectRatio -> Automatic), Frame -> True, FrameTicks -> None, GridLines -> None]; wt = {{1,1,1}, {1,9,1}, {1,1,1}}; rule= 261634; init={{1}}; Show[GraphicsArray[Map[RasterGraphics, CellularAutomaton[{rule, {2, wt}, {1, 1}}, {init, 0}, 9, -10]]]];nx = 100; ca = CellularAutomaton[{rule, {2, wt}, {1, 1}}, {init, 0}, nx - 1, -nx]; a = Table[Total[ca[[i]], 2], {i, 1, nx}] (* John W. Layman, Sep 11 2009 *)
A151725[0] = 0; A151725[n_] := Total[CellularAutomaton[{174766, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, {{{n - 1}}}], 2]; Array[A151725, 48, 0] (* JungHwan Min, Sep 01 2016 *)
A151725L[n_] := Prepend[Total[#, 2] & /@ CellularAutomaton[{174766, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, n - 1], 0]; A151725L[47] (* JungHwan Min, Sep 01 2016 *)

For a recurrence see the Applegate-Pol-Sloane paper.

Definition clarified by SiYang Hu, May 10 2025

A161644 Number of ON states after n generations of cellular automaton based on triangles.

Original entry on oeis.org

0, 1, 4, 10, 16, 22, 34, 52, 64, 70, 82, 106, 136, 160, 190, 232, 256, 262, 274, 298, 328, 358, 400, 466, 532, 568, 598, 658, 742, 814, 892, 988, 1036, 1042, 1054, 1078, 1108, 1138, 1180, 1246, 1312, 1354, 1396, 1474, 1588, 1702, 1816, 1966, 2104, 2164, 2194

Offset: 0

David Applegate and N. J. A. Sloane, Jun 15 2009

nonn

Analog of A151723 and A151725, but here we are working on the hexagonal net where each triangular cell has three neighbors (meeting along its edges). A cell is turned ON if exactly one of its three neighbors is ON. An ON cell remains ON forever.
We start with a single ON cell.
There is a dual version where the triangular cells meet vertex-to-vertex. The counts are the same: the two versions are isomorphic. Reed (1974) uses the vertex-to-vertex version. See the two Sloane "Illustration" links below to compare the two versions.
It appears that a(n) is also the number of polytoothpicks added in a toothpick structure formed by V-toothpicks but starting with a Y-toothpick: a(n) = a(n-1)+(A182632(n)-A182632(n-1))/2. (Checked up to n=39.) - Omar E. Pol, Dec 07 2010 and R. J. Mathar, Dec 17 2010
It appears that the behavior is similar to A161206. - Omar E. Pol, Jan 15 2016
It would be nice to have a formula or recurrence.
If new triangles are required to always move outwards we get A295559 and A295560.
From Paul Cousin, May 23 2025: (Start)
This is ETA rule 242 (11110010 in binary):
  -----------------------------------------------
  |state of the cell            |1|1|1|1|0|0|0|0|
  |sum of the neighbors' states |3|2|1|0|3|2|1|0|
  |cell's next state            |1|1|1|1|0|0|1|0|
  ----------------------------------------------- (End)

R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Describes the dual structure where new triangles are joined at vertices rather than edges.]
S. 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 3.

Paul Cousin, Table of n, a(n) for n = 0..16385 (first 10001 terms from Rémy Sigrist)
David Applegate, The movie version
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.]
Paul Cousin, ETA rule 242
Paul Cousin, Triangular Automata Integer Sequences
Lucas Garron, first 64 steps
Lucas Garron, after 128 steps
R. Reed, The Lemming Simulation Problem, Mathematics in School, 3 (#6, Nov. 1974), front cover and pp. 5-6. [Scanned photocopy of pages 5, 6 only, with annotations by R. K. Guy and N. J. A. Sloane]
Rémy Sigrist, PARI program for A161644
N. J. A. Sloane, Illustration of first 7 generations of A161644 and A295560 (edge-to-edge version)
N. J. A. Sloane, Illustration of first 11 generations of A161644 and A295560 (vertex-to-vertex version) [Include the 6 cells marked x to get A161644(11), exclude them to get A295560(11).]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Cf. A151723, A151725, A147562, A161206, A161645, A139250, A160120, A161206, A182632, A182840, A250300, A295559, A295560.

PARI
```
\\ See Links section.
```

a(n) = (A182632(n) - 1)/2, n >= 1. - Omar E. Pol, Mar 07 2013

Edited by N. J. A. Sloane, Jan 10 2010 and Nov 27 2017

A182632 Toothpick sequence on the hexagonal net starting from a node.

Original entry on oeis.org

0, 3, 9, 21, 33, 45, 69, 105, 129, 141, 165, 213, 273, 321, 381, 465, 513, 525, 549, 597, 657, 717, 801, 933, 1065, 1137, 1197, 1317, 1485, 1629, 1785, 1977, 2073, 2085, 2109, 2157, 2217, 2277, 2361, 2493, 2625, 2709, 2793, 2949, 3177, 3405, 3633

Offset: 0

Omar E. Pol, Dec 07 2010

nonn

A connected network of toothpicks is constructed by the following iterative procedure. At stage 1, place three toothpicks each of length 1 on a hexagonal net, as a propeller, joined at a node. At each subsequent stage, add two toothpicks (which could be called a single V-toothpick with a 120-degree corner) adjacent to each node which is the endpoint of a single toothpick.
The exposed endpoints of the toothpicks of the old generation are touched by the endpoints of the toothpicks of the new generation. In the graph, the edges of the hexagons become edges of the graph, and the graph grows such that the nodes that were 1-connected in the old generation are 3-connected in the new generation.
It turns out heuristically that this growth does not show frustration, i.e., a free edge is never claimed by two adjacent exposed endpoints at the same stage; the rule of growing the network does apparently not need specifications to address such cases.
The sequence gives the number of toothpicks in the toothpick structure after n-th stage. A182633 (the first differences) gives the number of toothpicks added at n-th stage.
a(n) is also the number of components after n-th stage in a toothpick structure starting with a single Y-toothpick in stage 1 and adding only V-toothpicks in stages >= 2. For example: consider that in A161644 a V-toothpick is also a polytoothpick with two components or toothpicks and a Y-toothpick is also a polytoothpick with three components or toothpicks. For more information about this comment see A161206, A160120 and A161644.
Has a behavior similar to A151723, A182840. - Omar E. Pol, Mar 07 2013
From Omar E. Pol, Feb 17 2023: (Start)
Assume that every triangular cell has area 1.
It appears that the structure contains only three types of polygons:
- Regular hexagons of area 6.
- Concave decagons (or concave 10-gons) of area 12.
- Concave dodecagons (or concave 12-gons) of area 18.
There are infinitely many of these polygons.
The structure contains concentric hexagonal rings formed by hexagons and also contains concentric hexagonal rings formed by alternating decagons and dodecagons.
The structure has internal growth.
For an animation see the movie version in the Links section.
The animation shows the fractal-like behavior the same as in other members of the family of toothpick cellular automata.
For another version starting with a simple toothpick see A182840.
For a version of the structure in the first quadrant but on the square grid see A182838. (End)

			a(0)=0. At stage 1 we place 3 toothpicks connected to the initial grid point of the structure. Note that there are 3 exposed endpoints. At stage 2 we place 6 toothpicks, so a(2)=3+6=9, etc.

David Applegate, The movie version
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
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

Cf. A139250, A160120, A161206, A161644, A182617, A182633, A182634, A182836, A182838, A182840.

a(n) = 3*A182634(n).

a(n) = 1 + 2*A161644(n), n >= 1. - Omar E. Pol, Mar 07 2013

A169779 Partial sums of A170898.

Original entry on oeis.org

1, 2, 5, 6, 9, 14, 21, 22, 25, 30, 39, 48, 55, 68, 83, 84, 87, 92, 101, 110, 119, 136, 161, 178, 185, 198, 221, 248, 267, 298, 329, 330, 333, 338, 347, 356, 365, 382, 407, 424, 433, 450, 479, 516, 549, 590, 647, 680, 687, 700, 723, 750, 777, 820, 887, 946, 973, 1004, 1059

Offset: 0

N. J. A. Sloane, May 10 2010

nonn

On the hexagonal grid consider an infinite 60-degree wedge. A cell is turned ON if exactly one of its six neighbors is ON. We start with a single ON cell. An ON cell remains ON forever. The sequence counts the total number of ON states after n generations. The structure is also the tree that arises from one of the six spokes of the structure of A151723. For n >> 1 the structure looks like a quadrilateral formed by two scalene right triangles which are joined at their hypotenuses. - Omar E. Pol, Mar 06 2013

N. J. A. Sloane, Table of n, a(n) for n = 0..1023
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A151723, A169780.

a(n) = (A151723(n+2) - 1)/6. - Omar E. Pol, Mar 06 2013

A170898 Triangle read by rows, obtained by dividing A151724 by 6.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 7, 1, 3, 5, 9, 9, 7, 13, 15, 1, 3, 5, 9, 9, 9, 17, 25, 17, 7, 13, 23, 27, 19, 31, 31, 1, 3, 5, 9, 9, 9, 17, 25, 17, 9, 17, 29, 37, 33, 41, 57, 33, 7, 13, 23, 27, 27, 43, 67, 59, 27, 31, 55, 69, 49, 69, 63, 1, 3, 5, 9, 9, 9, 17, 25, 17, 9, 17, 29, 37, 33, 41

Offset: 0

N. J. A. Sloane, Jan 10 2010

Row k has 2^k terms.

Right border gives the positive terms of A000225. - Omar E. Pol, Sep 28 2013

			Triangle begins:
1;
1,3;
1,3,5,7;
1,3,5,9,9,7,13,15;
1,3,5,9,9,9,17,25,17,7,13,23,27,19,31,31;
1,3,5,9,9,9,17,25,17,9,17,29,37,33,41,57,33,7,13,23,27,27,43,67,59,27,31,55,69,49,69,63;
...

N. J. A. Sloane, Table of n, a(n) for n = 0..1023

Cf. A169779 (partial sums).

Cf. A139250, A139251, A151723, A151724, A170899.

Equals A170905(n) - 1.

A170899 Triangle read by rows, obtained by subtracting 1 from the terms of A170898 and dividing by 2.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 0, 1, 2, 4, 4, 3, 6, 7, 0, 1, 2, 4, 4, 4, 8, 12, 8, 3, 6, 11, 13, 9, 15, 15, 0, 1, 2, 4, 4, 4, 8, 12, 8, 4, 8, 14, 18, 16, 20, 28, 16, 3, 6, 11, 13, 13, 21, 33, 29, 13, 15, 27, 34, 24, 34, 31, 0, 1, 2, 4, 4, 4, 8, 12, 8, 4, 8, 14, 18, 16, 20, 28, 16, 4, 8, 14

Offset: 0

N. J. A. Sloane, Jan 10 2010

This sequence is essentially the number of cells that are turned ON at the n-th generation of a 30-degree sector of the hexagonal Ulam-Warburton cellular automaton in A151723. The cells on the six main diagonals are ignored, and the resulting counts have been divided by 12. - N. J. A. Sloane, Mar 13 2021
Row k has 2^k terms.
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. - N. J. A. Sloane, Mar 14 2021
It appears that this may also be regarded as a tetrahedron E(m,i,j), m>=0, i>=0, j>=0, in which the slice m is a triangle read by rows: R(i,j) in which row i has length A011782(i). - Omar E. Pol, Feb 13 2013
It appears that in the slice m (of the tetrahedron mentioned above) the differences between the first 2^(m-3) elements of row m-1 and the first 2^(m-3) elements of row m give the first 2^(m-3) elements of A169787, if m >= 3. Also it appears that the right border of slice m gives the first m powers of 2 together with 0. See the second arrangement in Example section. - Omar E. Pol, Mar 16 2013

			Triangle begins:
0;
0,1;
0,1,2,3;
0,1,2,4,4,3,6,7;
0,1,2,4,4,4,8,12,8,3,6,11,13,9,15,15;
0,1,2,4,4,4,8,12,8,4,8,14,18,16,20,28,16,3,6,11,13,13,21, 33,29,13,15,27,34,24,34,31;
0,1,2,4,4,4,8,12,8,4,8,14,18,16,20,28,16,4,8,14,18,18,26,42,42,24,20,36,50,46,50,62,32,3,6,11,13,13,21,33,29,17,21, 37,51,51,57,77,61,21,15,27,34,36,52,80,80,44,38,62,81,58, 73,63;
0,1,2,4,4,4,8,12,8,4,8,14,18,16,20,28,16,4,8,14,18,18,26,42,42,24,20,36,50,46,50,62,32,4,8,14,18,18,26,42,42,26,26, 46,66,70,74,98,90,40,20,36,50,54,70,110,126,86,58,86,124, 118,118,132,64,3,6,11,13,13,21,33,29,17,21,37,51,51,57,77,61,25,21,37,51,55,71,111,127,91,65,93,137,143,147,175,127,37,15,27,34,36,52,80,80,56,56,88,126,136,150,192,172,84,46,62,81,90,124,184,196,124,96,139,183,131,152,127;
...
From _Omar E. Pol_, Feb 13 2013 (Start):
When written as a tetrahedron the slices 0-7 are:
0;
..
1;
0;
..
1;
2;
3,0;
....
1;
2;
4,4;
3,6,7,0;
........
1;
2;
4,4;
4,8,12,8;
3,6,11,13,9,15,15,0;
....................
1;
2;
4,4;
4,8,12,8;
4,8,14,18,16,20,28,16;
3,6,11,13,13,21,33,29,13,15,27,34,24,34,31,0;
.............................................
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;
..........................................................
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;
4,8,14,18,18,26,42,42,26,26,46,66,70,74,98,90,40,20,36,50,54,70,110,126,86,58,86,124,118,118,132,64;
3,6,11,13,13,21,33,29,17,21,37,51,51,57,77,61,25,21,37,51,55,71,111,127,91,65,93,137,143,147,175,127,37,15,27,34,36,52,80,80,56,56,88,126,136,150,192,172,84,46,62,81,90,124,184,196,124,96,139,183,131,152,127,0;
..........................................................
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..4092

Cf. A139250, A151723, A151724, A170898.

A342272, A342273, A342274 are limiting sequences to which various parts of the rows of this triangle converge.

A342272 The rows of the triangle A170899 converge to this sequence.

Original entry on oeis.org

0, 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, 4, 8, 14, 18, 18, 26, 42, 42, 26, 26, 46, 66, 70, 74, 98, 90, 40, 20, 36, 50, 54, 70, 110, 126, 86, 58, 86, 124, 118, 118, 132

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.
A342273 is the limiting sequence for the part of the row of A170899 that start at the first "3".
Needs a bigger b-file.

N. J. A. Sloane, Table of n, a(n) for n = 0..512

Cf. A151723, A151724, A170899, A342273, A342274.

This is A169787 with 1 subtracted from each term.

A342278 First differences of A342274.

Original entry on oeis.org

4, 6, 4, 0, 8, 16, 0, -16, 0, 20, 20, 4, 4, 24, -8, -48, -16, 20, 20, 8, 16, 48, 32, -36, -44, 24, 60, 20, 0, 32, -36, -116, -48, 20, 20, 8, 16, 48, 32, -32, -32, 40, 80, 48, 16, 56, 32, -116, -140, -8, 60, 36, 40, 112, 112, -48, -136, 0, 132, 64, -12, 28, -108, -256, -112, 20, 20, 8, 16, 48, 32

Offset: 0

N. J. A. Sloane, Mar 14 2021

sign

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.

N. J. A. Sloane, Table of n, a(n) for n = 0..254

Cf. A151723, A342274.

cp's OEIS Frontend

A334228 a(n) is the n-th completed odd-numbered level 2k-1 containing k ON-cells in a triangular wedge in the hexagonal grid of A151723 (i.e., after 2^j >= 2k-1 generations of the automaton in A151723 have been computed).

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A182840 Toothpick sequence on hexagonal net.

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A151725 Number of ON states after n generations of cellular automaton rule described by the rulestring B1/S012345678.

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A161644 Number of ON states after n generations of cellular automaton based on triangles.

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PARI

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A182632 Toothpick sequence on the hexagonal net starting from a node.

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A169779 Partial sums of A170898.

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A170898 Triangle read by rows, obtained by dividing A151724 by 6.

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A170899 Triangle read by rows, obtained by subtracting 1 from the terms of A170898 and dividing by 2.

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A342272 The rows of the triangle A170899 converge to this sequence.

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