A147562 -id:A147562 - cp's OEIS frontend

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

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

A151920 a(n) = (Sum_{i=1..n+1} 3^wt(i))/3, where wt() = A000120().

Original entry on oeis.org

1, 2, 5, 6, 9, 12, 21, 22, 25, 28, 37, 40, 49, 58, 85, 86, 89, 92, 101, 104, 113, 122, 149, 152, 161, 170, 197, 206, 233, 260, 341, 342, 345, 348, 357, 360, 369, 378, 405, 408, 417, 426, 453, 462, 489, 516, 597, 600, 609, 618, 645, 654, 681, 708, 789, 798, 825, 852, 933, 960

Offset: 0

N. J. A. Sloane, Aug 05 2009, Aug 06 2009

Partial sums of A147610 (but with offset changed to 0).

It appears that the first bisection gives the positive terms of A147562. - Omar E. Pol, Mar 07 2015

			n=3: (3^1+3^1+3^2+3^1)/3 = 18/3 = 6.
n=18: the binary expansion of 18+1 is 10011, i.e., 19 = 2^4 + 2^1 + 2^0.
The exponents of these powers of 2 (4, 1 and 0) reoccur as exponents in the powers of 4: a(19) = 3^0 * [(4^4 - 1) / 3 + 1] + 3^1 * [(4^1 - 1) / 3 + 1] + 3^2 * [(4^0 - 1)/3 + 1] = 1 * 86 + 3 * 2 + 9 * 1 = 101. - _David A. Corneth_, Mar 21 2015

Michael De Vlieger, Table of n, a(n) for n = 0..10000
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. 31.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000120, A130665, A147562, A147610, A153000, A160410, A160414.

Cf. A139250, A151917, A152998.

t = Nest[Join[#, # + 1] &, {0}, 14]; Table[Sum[3^t[[i + 1]], {i, 1, n}]/3, {n, 60}] (* Michael De Vlieger, Mar 21 2015 *)

a(n) = sum(i=1, n+1, 3^hammingweight(i))/3; \\ Michel Marcus, Mar 07 2015

a(n)={b=binary(n+1);t=#b;e=-1;sum(i=1,#b,e+=(b[i]==1);(b[i]==1)*3^e*((4^(#b-i)-1)/3+1))} \\ David A. Corneth, Mar 21 2015

a(n) = (A147562(n+2) - 1)/4 = (A151917(n+2) - 1)/2. - Omar E. Pol, Mar 13 2011
a(n) = (A130665(n+1) - 1)/3. - Omar E. Pol, Mar 07 2015
a(n) = a(n-1) + 3^A000120(n+1)/3. - David A. Corneth, Mar 21 2015

A160170 X-toothpick sequence on Z^3 lattice (see Comments for precise definition).

Original entry on oeis.org

0, 1, 5, 13, 21, 45, 77, 109, 165, 245, 325, 413, 525, 685, 853, 1093, 1317, 1661, 1981, 2301, 2645, 3093, 3621, 4157, 4861, 5565, 6437, 7173, 8053, 8893, 9917, 11005, 12261, 13589, 14981, 16397, 17837, 19341, 20997

Offset: 0

Omar E. Pol, May 03 2009

nonn

Here a "X-toothpick" is defined to be a cross with 4 endpoints and a midpoint. Also, a X-toothpick can be represented by set of four connected toothpicks forming a cross.
We start at stage 0 on the Z^3 lattice with no X-toothpicks.
At stage 1 place a X-toothpick.
Rule: each exposed endpoint of the X-toothpicks of the old generation must be touched by the midpoint of a X-toothpick of new generation (see illustrations).
The sequence gives the number of X-toothpicks in the three-dimensional structure after n-th stage. A170171 (the first differences) gives the number of X-toothpicks added at n-th stage.
For a similar sequence but starting with a single toothpick see A170876.
For the X-toothpick sequence on Z^2 lattice see A147562, the Ulam-Warburton cellular automaton.
For more information about the growth of toothpicks see A139250.

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.],
R. J. Mathar, C++ program
R. J. Mathar, Views after stages 1 to 10 (PDF)
Olbaid Fractalium, Toothpick sequence Part 1, (Watch from minute 3:14), Youtube video (2023).
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to toothpick sequences

Cf. A139250, A160120, A160160, A160171, A170876.

Cf. A147562. - Omar E. Pol, Mar 28 2011

C++ program, illustrations and more terms (a(6)-a(38)) based on Email from R. J. Mathar dated Jan 10 2010.

A160117 Number of "ON" cells after n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Original entry on oeis.org

0, 1, 9, 13, 41, 49, 101, 113, 189, 205, 305, 325, 449, 473, 621, 649, 821, 853, 1049, 1085, 1305, 1345, 1589, 1633, 1901, 1949, 2241, 2293, 2609, 2665, 3005, 3065, 3429, 3493, 3881, 3949, 4361, 4433, 4869, 4945, 5405, 5485, 5969, 6053, 6561, 6649, 7181, 7273

Offset: 0

Omar E. Pol, May 05 2009, May 15 2009

nonn

Define "peninsula cell" to be the "ON" cell connected to the structure by exactly one of its vertices.
Define "bridge cell" to be the "ON" cell connected to two cells of the structure by exactly consecutive two of its vertices.
On the infinite square grid, we start at stage 0 with all cells in OFF state. At stage 1, we turn ON a single cell, in the central position.
In order to construct this sequence we use the following rules:
- If n is even, we turn "ON" the cells around the cells turned "ON" at the generation n-1.
- If n is odd, we turn "ON" the possible bridge cells and the possible peninsula cells.
- Everything that is already ON remains ON.
A160411, the first differences, gives the number of cells turned "ON" at n-th stage.

			If we label the generations of cells turned ON by consecutive numbers we get the cell pattern shown below:
9...9...9...9...9
.888.888.888.888.
.878.878.878.878.
.886668666866688.
9..656.656.656..9
.886644464446688.
.878.434.434.878.
.886644222446688.
9..656.212.656..9
.886644222446688.
.878.434.434.878.
.886644464446688.
9..656.656.656..9
.886668666866688.
.878.878.878.878.
.888.888.888.888.
9...9...9...9...9
At the first generation, only the central "1" is ON, so a(1) = 1. At the second generation, we turn ON eight cells around the central cell, leading to a(2) = a(1)+8 = 9. At the third generation, we turn ON four peninsula cells, so a(3) = a(2)+4 = 13. At the fourth generation, we turn ON the cells around the cells turned ON at the third generation, so a(4) = a(3)+28 = 41. At the 5th generation, we turn ON four peninsula cells and four bridge cells, so a(5) = a(4)+8 = 49.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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 cellular automata

Cf. A139250, A139251, A147562, A160118, A160119, A160379.

a:= proc(n) local r;
      r:= irem(n, 2);
      `if`(n<2, n, 5+(n-r)*((7*n-3*r)/2-5))
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, Sep 16 2011

a[0] = 0; a[1] = 1; a[n_] := If[EvenQ[n], (7n^2 - 10n + 10)/2, (7n^2 - 20n + 23)/2]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jul 16 2015, after Nathaniel Johnston *)

a(2n) = 5 + 2n(7n-5) for n>=1, a(2n+1) = 5 + 2n(7n-3) for n>=1. - Nathaniel Johnston, Nov 06 2010

G.f.: x*(x^2+1)*(4*x^3+x^2+8*x+1)/((x+1)^2*(1-x)^3). - Alois P. Heinz, Sep 16 2011

a(10) - a(27) from Nathaniel Johnston, Nov 06 2010

a(28) - a(47) from Alois P. Heinz, Sep 16 2011

A151779 a(1)=1; for n > 1, a(n)=6*5^{wt(n-1)-1}.

Original entry on oeis.org

1, 6, 6, 30, 6, 30, 30, 150, 6, 30, 30, 150, 30, 150, 150, 750, 6, 30, 30, 150, 30, 150, 150, 750, 30, 150, 150, 750, 150, 750, 750, 3750, 6, 30, 30, 150, 30, 150, 150, 750, 30, 150, 150, 750, 150, 750, 750, 3750, 30, 150, 150, 750, 150, 750, 750, 3750, 150, 750, 750, 3750

Offset: 1

N. J. A. Sloane, Jun 25 2009

Number of cells turned ON in n-th generation of cellular automaton based on Z^3 lattice in the same way that A147562 is based on the Z^2 lattice. Here each cell has six neighbors.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
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. A147582, A151780, A151781, A151782.

wt := proc(n) local w,m,i; w := 0; m := n; while m > 0 do i := m mod 2; w := w+i; m := (m-i)/2; od; w; end:
f:=d->[seq((2*d)*(2*d-1)^(wt(n-1)-1),n=2..120)];
f2:=d->[1,op(f(d))];
f2(3);

a(n)=6*5^(hammingweight(n-1)-1)\1 \\ Charles R Greathouse IV, Mar 07 2015

A160420 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton whose skeleton is the same network as the toothpick structure of A139250 but with toothpicks of length 4.

Original entry on oeis.org

0, 5, 13, 27, 41, 57, 85, 123, 149, 165, 193, 233, 277, 337, 429, 527, 577, 593, 621, 661, 705, 765, 857, 957, 1025, 1085, 1181, 1305, 1453, 1665, 1945, 2187, 2285, 2301, 2329, 2369, 2413, 2473, 2565, 2665, 2733, 2793, 2889, 3013, 3161, 3373, 3653, 3897, 4013

Offset: 0

Omar E. Pol, May 13 2009, May 18 2009

nonn

a(n) is also the number of grid points that are covered after n-th stage by an polyedge as the toothpick structure of A139250, but with toothpicks of length 4.

			a(2)=13:
.o-o-o-o-o
.....|....
.....o....
.....|....
.....o....
.....|....
.....o....
.....|....
.o-o-o-o-o

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. A139250, A139251, A147614, A147562, A160118, A160120, A160170, A160430.

Conjecture: a(n) = A147614(n)+2*A139250(n). [From R. J. Mathar, Jan 22 2010]

The above conjecture is true: each toothpick covers exactly two more grid points than the corresponding toothpick in A147614.

Definition revised by N. J. A. Sloane, Jan 02 2010.

Formula verified and more terms from Nathaniel Johnston, Nov 13 2010

A255741 Square array read by antidiagonals upwards: T(n,k), n>=1, k>=1, in which row n lists the partial sums of the n-th row of the square array of A255740.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 5, 3, 1, 1, 5, 7, 7, 4, 1, 1, 6, 9, 13, 9, 4, 1, 1, 7, 11, 21, 16, 11, 4, 1, 1, 8, 13, 31, 25, 22, 13, 4, 1, 1, 9, 15, 43, 36, 37, 28, 15, 5, 1, 1, 10, 17, 57, 49, 56, 49, 40, 17, 5, 1, 1, 11, 19, 73, 64, 79, 76, 85, 43, 19, 5, 1, 1, 12, 21, 91, 81, 106, 109, 156, 89, 49, 21, 5, 1

Offset: 1

Omar E. Pol, Mar 05 2015

			The corner of the square array with the first 15 terms of the first 12 rows looks like this:
-------------------------------------------------------------------------
A000012: 1, 1, 1,  1,  1,  1,  1,   1,   1,   1,   1,   1,   1,   1,   1
A070941: 1, 2, 3,  3,  4,  4,  4,   4,   5,   5,   5,   5,   5,   5,   5
A005408: 1, 3, 5,  7,  9, 11, 13,  15,  17,  19,  21,  23,  25,  27,  29
A151788: 1, 4, 7, 13, 16, 22, 28,  40,  43,  49,  55,  67,  73,  85,  97
A147562: 1, 5, 9, 21, 25, 37, 49,  85,  89, 101, 113, 149, 161, 197, 233
A151790: 1, 6,11, 31, 36, 56, 76, 156, 161, 181, 201, 281, 301, 381, 461
A151781: 1, 7,13, 43, 49, 79,109, 259, 265, 295, 325, 475, 505, 655, 805
A151792: 1, 8,15, 57, 64,106,148, 400, 407, 449, 491, 743, 785,1037,1289
A151793: 1, 9,17, 73, 81,137,193, 585, 593, 649, 705,1097,1153,1545,1937
A255764: 1,10,19, 91,100,172,244, 820, 829, 901, 973,1549,1621,2197,2773
A255765: 1,11,21,111,121,211,301,1111,1121,1211,1301,2111,2201,3011,3821
A255766: 1,12,23,133,144,254,364,1464,1475,1585,1695,2795,2905,4005,5105
...

Index entries for sequences related to cellular automata

Rows 1-12: A000012, A070941, A005408, A151788, A147562, A151790, A151781, A151792, A151793, A255764, A255765, A255766.
Columns 1-10: A000012, A000027, A005408, A002061, A000290, A084849, A056107, A053698, A100705, A100104.
Cf. A000120, A255740.

A319018 Number of ON cells after n generations of two-dimensional automaton based on knight moves (see Comments for definition).

Original entry on oeis.org

0, 1, 9, 17, 57, 65, 121, 145, 265, 273, 329, 377, 617, 657, 865, 921, 1201, 1209, 1265, 1313, 1553, 1617, 2001, 2121, 2689, 2745, 3009, 3153, 3841, 3953, 4513, 4649, 5297, 5305, 5361, 5409, 5649, 5713, 6097, 6233, 6881, 6953, 7353, 7585, 8713, 8913, 9961

Offset: 0

Rémy Sigrist, Sep 08 2018

nonn

The cells are the squares of the standard square grid.
Cells are either OFF or ON, once they are ON they stay ON forever.
Each cell has 8 neighbors, the cells that are a knight's move away.
We begin in generation 1 with a single ON cell.
A cell is turned ON at generation n+1 if it has exactly one ON neighbor at generation n.
(Since cells stay ON, an equivalent definition is that a cell is turned ON at generation n+1 if it has exactly one neighbor that has been turned ON at some earlier generation. - N. J. A. Sloane, Dec 19 2018)
This sequence has similarities with A151725: here we use knight moves, there we use king moves.
This is a knight's-move version of the Ulam-Warburton cellular automaton (see A147562). - N. J. A. Sloane, Dec 21 2018
The structure has dihedral D_8 symmetry (quarter-turn rotations plus reflections, which generate the dihedral group D_8 of order 8), so A319019 is a multiple of 8 (compare A322050). - N. J. A. Sloane, Dec 16 2018
From Omar E. Pol, Dec 16 2018: (Start)
For n >> 1 (for example: n = 257) the structure of this sequence is similar to the structure of both A194270 and of A220500, the D-toothpick cellular automata of the second kind and of the third kind respectively. The animations of both CAs are in the Applegate's movie version.
Also, the graph of A319018 is a bit similar to the graph of A245540, which is essentially a 45-degree-3D-wedge of A245542 (a pyramid) which is the partial sums of A160239 (Fredkin's replicator). See "Plot 2": A319018 vs. A245540. (End)
The conjecture that A322050(2^k+1)=1 also suggests a fractal geometry. Let P_k be the associated set of eight points. It appears that P_k may be written as the intersection of four fixed lines, y = +-2*x and x = +-2*y, with a circle, x^2 + y^2 = 5*4^k (see linked image "Log-Periodic Coloring"). - Bradley Klee, Dec 16 2018
In many of these toothpick or cellular automata sequences it is common to see graphs which look like some version of the famous blancmange curve (also known as the Takagi curve). I expect that is what we are seeing when we look at the graph of A322049, although we probably need to go a lot further out before the true shape becomes apparent. - N. J. A. Sloane, Dec 17 2018
The graph of A322049 (related to first differences of this sequence) appears to have rather a self-similar structure which repeats at powers of 2, and more specifically at 2^10 = 1024. There is no central symmetry or continuity, which are characteristic properties of the blancmange curve. - M. F. Hasler, Dec 28 2018
The 8 points added in generation n = 2^k + 1 are P_k = 2^k*K where K = {(+-2, +-1), (+-1, +-2)} is the set of the initial 8 knight moves. So P_k is indeed the intersection of the rays of slope +-1/2 resp. +-2 and a circle of radius 2^k*sqrt(5). In the subsequent generation n = 2^k + 2, the new cells switched on are exactly the 7 "new" knight move neighbors of these 8 cells, (P_k + K) \ (2^k - 1)*K. The 8th neighbor, lying one knight move closer to the origin, has been switched on in generation 2^k, together with an octagonal "wall" consisting of every other cell on horizontal and vertical segments between these points (2^k - 1)*K, and all cells on the diagonal segments between these points, as well as 2 more diagonals just next to these (on the inner side) and shorter by 2 cells (so they are empty for k = 1). This yields 4*(2 + (2^k - 2)*(1+3)) new ON cells in generation 2^k, plus 8*(2^(k-1) - 2) more new ON cells on horizontal, vertical and diagonal lines 4 units closer to the origin for k > 2, and similar additional terms for k > 4 etc. - M. F. Hasler, Dec 28 2018

Rémy Sigrist, Table of n, a(n) for n = 0..2049
David Applegate, The movie version.
Nathan Epstein, Gfycat animation of sequence.
M. F. Hasler, Interactive illustration of A319018 and A319019, Dec. 2018.
Bradley Klee, Log-Periodic Coloring at stage 257.
Bradley Klee, Log-periodic coloring of the first quadrant, over the chair tiling.
Rémy Sigrist, Illustration of the structure at stage 7
Rémy Sigrist, Illustration of the structure at stage 257
Rémy Sigrist, Colored illustration of the structure at stage 257 (where the hue is a function of the stage)
Rémy Sigrist, PARI program for A319018
N. J. A. Sloane, Hand-drawn sketch showing terms through about the eighth shell, but using offset a(0)=1. Illustrates the octagonal "castle walls".
N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)
Index entries for sequences related to cellular automata
Index entries for sequences related to toothpick sequences

Cf. A151725, A319019 (first differences).
For further analysis see A322048, A322049, A322050, A322051.
See A322055, A322056 for a variation.
See also A139250, A147562, A194270, A220500.

A319018(n)=sum(i=1,n,A319019[i]) \\ with array A319019=A319019_upto(N) precomputed with sufficiently large N. - M. F. Hasler, Dec 28 2018

No formula or recurrence is presently known. See A322049 for a promising attack. - N. J. A. Sloane, Dec 16 2018

a(n) = Sum_{k=1..n} A319019(n) = 1 + 8*Sum_{k=2..n} A322050(n) for n >= 1. In particular, a(n) - 1 is divisible by 8 for all n >= 1. - M. F. Hasler, Dec 28 2018

Deleted an incorrect illustration. - N. J. A. Sloane, Dec 17 2018

A323650 Flower garden sequence (see Comments for precise definition).

Original entry on oeis.org

0, 1, 3, 7, 15, 19, 27, 39, 63, 67, 75, 87, 111, 123, 147, 183, 255, 259, 267, 279, 303, 315, 339, 375, 447, 459, 483, 519, 591, 627, 699, 807, 1023, 1027, 1035, 1047, 1071, 1083, 1107, 1143, 1215, 1227, 1251, 1287, 1359, 1395, 1467, 1575, 1791, 1803, 1827, 1863, 1935, 1971, 2043, 2151, 2367, 2403, 2475

Offset: 0

Omar E. Pol, Jan 21 2019

nonn

This arises from a hybrid cellular automaton on a triangular grid formed of I-toothpicks and V-toothpicks. Also, it appears that this is a missing link between A147562 (Ulam-Warburton) and three toothpick sequences: A139250 (normal toothpicks), A161206 (V-toothpicks) and A160120 (Y-toothpicks). The behavior resembles the toothpick sequence A139250, on the other hand, the formulas are directly related to A147562. Plot 2 shows that the graph is located between the graph of A139250 and the graph of A147562.
For the construction of the sequence the rules are as follows:
On the infinite triangular grid at stage 0 there are no toothpicks, so a(0) = 0.
At stage 1 we place an I-toothpick formed of two single toothpicks in vertical position, so a(1) = 1.
For the next n generation we have that:
If n is even then at every free end of the structure we add a V-toothpick formed of two single toothpicks such that the angle of each single toothpick with respect to the connected I-toothpick is 120 degrees.
If n is odd then we add I-toothpicks in vertical position (see the example).
a(n) gives the total number of I-toothpicks and V-toothpicks in the structure after the n-th stage.
A323651 (the first differences) gives the number of elements added at the n-th stage.
Note that 2*a(n) gives the total number of single toothpicks of length 1 after the n-th stage.
The structure contains only three kinds of polygonal regions as follows:
- Rhombuses that contain two triangular cells.
- Regular hexagons that contain six triangular cells.
- Oblong hexagons that contain 10 triangular cells.
The structure looks like a "garden of flowers with six petals" (between other substructures). In particular, after 2^(n+1) stages with n >= 0, the structure looks like a flower garden in a rectangular box which contains A002450(n) flowers with six petals.
Note that this hybrid cellular automaton is also a superstructure of the Ulam-Warburton cellular automaton (at least in four ways). The explanation is as follows:
1) A147562(n) equals the total number of I-toothpicks in the structure after 2*n - 1 stage, n >= 1.
2) A147562(n) equals the total number of pairs of Y-toothpicks connected by their endpoints in the structure after 2*n stage (see the example).
3) A147562(n) equals the total number of "flowers with six petals" (or six-pointed stars formed of six rhombuses) in the structure after 4*n stage. Note that the location of the "flowers with six petals" in the structure is essentially the same as the location of the "ON" cells in the version "one-step bishop" of A147562.
4) For more connections to A147562 see the Formula section.
The "word" of this cellular automaton is "ab". For more information about the word of cellular automata see A296612.
The total number of “flowers with six petals” after n-th stage equals the total number of “hidden crosses” after n-th stage in the toothpick structure of A139250, including the central cross (beginning to count the crosses when their “nuclei” are totally formed with 4 quadrilaterals). - Omar E. Pol, Mar 06 2019

			Illustration of initial terms:
.
                        |   |
                \ /     |\ /|
         |       |        |
         |       |        |
                / \     |/ \|
                        |   |
n        1       2        3
a(n)     1       3        7
.
Note that for n = 2 the structure is also the same as a pair of Y-toothpicks connected by their endpoints (see A160120).

Cf. A002450, A103454, A139250 (normal toothpicks), A147562 (Ulam-Warburton), A147582, A160120 (Y-toothpicks), A161206 (V-toothpicks), A296612, A323641, A323642, A323649 (corner sequence), A323651 (first differences).

For other hybrid cellular automata, see A194270, A194700, A220500, A289840, A290220, A294020, A294962, A294980, A299770.

bw3[n_]:=bw3[n]=3^(DigitCount[n,2,1]-1);
A147562[n_]:=A147562[n]=If[n<2,n,1+4Sum[bw3[k],{k,n-1}]];
A323650[n_]:=If[OddQ[n],3A147562[(n-1)/2]+A147562[n]-A147562[n-1],3A147562[n/2]];
nterms=100;Array[A323650,nterms,0] (* Paolo Xausa, Jun 17 2022 *)

a(n) = 3*A147562(n/2) if n is even.
a(n) = 3*A147562((n-1)/2) + A147582(n) if n is odd.
a(n) = 3*A147562((n-1)/2) + A147562(n) - A147562(n-1) if n is odd.
a(2^n) = A103454(n), n >= 0.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

PARI

Formula

Extensions

A151920 a(n) = (Sum_{i=1..n+1} 3^wt(i))/3, where wt() = A000120().

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A160170 X-toothpick sequence on Z^3 lattice (see Comments for precise definition).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A160117 Number of "ON" cells after n-th stage in simple 2-dimensional cellular automaton (see Comments for precise definition).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A151779 a(1)=1; for n > 1, a(n)=6*5^{wt(n-1)-1}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

PARI

A160420 Number of "ON" cells at n-th stage in simple 2-dimensional cellular automaton whose skeleton is the same network as the toothpick structure of A139250 but with toothpicks of length 4.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A255741 Square array read by antidiagonals upwards: T(n,k), n>=1, k>=1, in which row n lists the partial sums of the n-th row of the square array of A255740.

Views