A151726 -id:A151726 - 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

A151731 Number of ON states after n generations of cellular automaton based on square grid with each cell adjacent to its eight neighbors.

Original entry on oeis.org

0, 2, 6, 14, 20, 32, 40, 54, 70, 94, 108, 128, 152, 172, 188, 224, 256, 300, 344, 380, 416, 464, 504, 552, 598, 658, 728, 772, 816, 880, 940, 1000, 1076, 1148, 1212, 1276, 1360, 1454, 1556, 1624, 1708, 1796, 1912, 2004, 2124, 2250, 2376, 2480

Offset: 0

N. J. A. Sloane, Jun 15 2009

nonn

A cell is turned ON if exactly two of its eight neighbors is ON. An ON cell remains ON forever.

We start with two edge-adjacent ON cells.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Rémy Sigrist, PARI program for A151731
Rémy Sigrist, Illustration of the structure at stage 512

See A151725, A151726 for the same CA except that exactly one neighbor must be ON for a cell to turn ON.

Cf. A151732, A151733, A151734.

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

A151747 Except for boundary cases (n <= 3, j = 0, 1, 2^i-1), satisfies a(n) = a(2^i+j) = 2 a(j) + a(j+1), where n = 2^i + j, 0 <= j < 2^i .

Original entry on oeis.org

0, 1, 3, 5, 8, 9, 11, 17, 21, 15, 11, 18, 25, 29, 39, 54, 53, 27, 11, 18, 25, 29, 39, 55, 57, 41, 40, 61, 79, 97, 132, 160, 129, 51, 11, 18, 25, 29, 39, 55, 57, 41, 40, 61, 79, 97, 132, 161, 133, 65, 40, 61, 79, 97, 133, 167, 155, 122, 141, 201, 255, 326, 424, 448, 305, 99, 11, 18

Offset: 0

David Applegate, Jun 16 2009

The boundary cases are covered by the following formulas:
a(n) = 2n-1 if n<=3.
a(n) = 1+(3*i+1)*2^(i-2) if j=0.
a(n) = 3+ 3*2^(i-1) if j= 1.
a(n) = 2*a(j)+a(j+1)-1 if j=2^i-1.

			If written as a triangle:
.0,
.1,
.3, 5,
.8, 9, 11, 17,
.21, 15, 11, 18, 25, 29, 39, 54,
.53, 27, 11, 18, 25, 29, 39, 55, 57, 41, 40, 61, 79, 97, 132, 160,
.129, 51, 11, 18, 25, 29, 39, 55, 57, 41, 40, 61, 79, 97, 132, 161, 133, 65, 40, 61, 79, 97, 133, 167, 155, 122, 141, 201, 255, 326, 424, 448,
.305, 99, 11, 18, 25, 29, 39, 55, 57, 41, 40, 61, 79, 97, 132, 161, 133, 65, 40, 61, 79, 97, 133, 167, 155, 122, 141, 201, 255, 326, 424, 449, 309, 113, 40, 61, 79, 97, 133, 167, 155, 122, 141, 201, 255, 326, 425, 455, 331, 170, 141, 201, 255, 327, 433, 489, 432, 385, 483, 657, 836, 1076, 1296, 1200,
.705, 195, 11, 18, 25, 29, 39, 55, 57, 41, 40, 61, 79, 97, 132, 161, 133, 65, 40, 61, 79, 97, 133, 167, 155, 122, 141, 201, 255, 326, 424, 449, 309, 113, 40, 61, 79, 97, 133, 167, 155, 122, 141, 201, 255, 326, 425, 455, 331, 170, 141, 201, 255, 327, 433, 489, 432, 385, 483, 657, 836, 1076, 1296, 1201, 709, 209, 40, 61, 79, 97, 133, 167, 155, 122, 141, 201, 255, 326, 425, 455, 331, 170, 141, 201, 255, 327, 433, 489, 432, 385, 483, 657, 836, 1076, 1297, 1207, 731, 266, 141, 201, 255, 327, 433, 489, 432, 385, 483, 657, 836, 1077, 1305, 1241, 832, 481, 483, 657, 837, 1087, 1355, 1410, 1249, 1253, 1623, ...
then the rows (omitting the first two terms of each row) converge to A151748.

David Applegate, The movie version [See A151725, variant]
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. A151725, A151726, A151735, A151748, A139250, A170879.

The first column gives A170881.

A151747 := proc(n) option remember; local i, j;
if (n <= 0) then
  0;
elif (n <= 3) then
  2*n-1;
else
   i := floor(log(n)/log(2));
   j := n - 2^i;
   if (j = 0) then (3*i+1)*2^(i-2)+1;
   elif (j = 1) then 3*2^(i-1)+3;
   elif (j = 2^i-1) then 2*procname(j)+procname(j+1)-1;
   else 2*procname(j)+procname(j+1);
   end if;
end if;
end proc;

a[n_] := a[n] = Module[{i, j}, Which[n <= 0, 0, n <= 3, 2n-1, True, i = Floor[Log2[n]]; j = n-2^i; Which[j == 0, (3i+1)*2^(i-2)+1, j == 1, 3*2^(i-1)+3, j == 2^i-1, 2a[j] + a[j+1] - 1,True, 2a[j] + a[j+1]]]];
Table[a[n], {n, 0, 67}] (* Jean-François Alcover, Aug 04 2022, from Maple code *)

cp's OEIS Frontend

A151727 If A151726 is written as a triangle, this is what the rows converge to.

Views

Author

Keywords

Crossrefs

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

A151731 Number of ON states after n generations of cellular automaton based on square grid with each cell adjacent to its eight neighbors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A151747 Except for boundary cases (n <= 3, j = 0, 1, 2^i-1), satisfies a(n) = a(2^i+j) = 2 a(j) + a(j+1), where n = 2^i + j, 0 <= j < 2^i .

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica