A160239 -id:A160239 - cp's OEIS frontend

A255477 Number of ON cells after n generations in 4-D cellular automaton defined by 4-dimensional analog of Moore neighborhood, when started with a single ON cell at generation 0.

1, 80, 80, 624, 80, 6400, 624, 16480, 80, 6400, 6400, 49920, 624, 49920, 16480, 221888, 80, 6400, 6400, 49920, 6400, 512000, 49920, 1318400, 624, 49920, 49920, 389376, 16480, 1318400, 221888, 4245888, 80, 6400, 6400, 49920, 6400, 512000, 49920, 1318400, 6400, 512000, 512000, 3993600, 49920, 3993600, 1318400

Offset: 0

Doron Zeilberger, Feb 26 2015

nonn

Run length transform of A255478.

Shalosh B. Ekhad, Details about A255477 and A255478
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796, 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249, 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
Index entries for sequences related to cellular automata

Cf. A255478.

See also A160239 and A246031 for analogs in 2 and 3 dimensions.

A275667 Number of ON cells after n generations in a 2-dimensional "Odd-Rule" cellular automaton on triangular tiling.

Original entry on oeis.org

1, 3, 7, 9, 7, 21, 25, 27, 7, 21, 49, 63, 25, 75, 103, 81, 7, 21, 49, 63, 49, 147, 175, 189, 25, 75, 175, 225, 103, 309, 409, 243, 7, 21, 49, 63, 49, 147, 175, 189, 49, 147, 343, 441, 175, 525, 721, 567, 25, 75, 175, 225, 175, 525, 625, 675, 103, 309, 721

Offset: 0

Kovba Alexey, Aug 04 2016

Each triangular tile has 3 neighbors. A cell is ON in a given generation if and only if there was an odd number of ON cells among the three nearest neighbors in the preceding generation.
At the initial moment there is a single ON cell.
Given pattern replicates after a number of generations which is a power of 2 when a(n) = 7.
Number of cells on each even step minus one is divisible by 6.
By analogy with the Ekhad, Sloane, Zeilberger link, one may suppose that using ternary expansion of n, recurrence relations for a(n) can be obtained and proved.
From Andrey Zabolotskiy, Aug 04 2016: (Start)
If the first conjecture from the Formula section is true then the fact that the right border of the triangle (see Example) gives A000244 follows directly from it.
If the second conjecture is true then the numbers just before the right border give A102900.
Since the 7 cells which are ON at the beginning of every row are farther and farther away from each other, the n-th term of a row (with offset 0) is a(n)*7 for not very large n.
See also comments to A247666.
(End)
This is ETA rule 170. See the Sadat-Benedek reference for proof of pattern replication. - Paul Cousin, Apr 22 2025

			From _Omar E. Pol_, Aug 04 2016: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
3;
7, 9;
7, 21, 25, 27;
7, 21, 49, 63, 25, 75, 103, 81;
7, 21, 49, 63, 49, 147, 175, 189, 25, 75, 175, 225, 103, 309, 409, 243;
...
It appears that the right border gives A000244.
(End)

Paul Cousin, Table of n, a(n) for n = 0..16384
Paul Cousin, Illustration for n = 0..128
Paul Cousin, Elementary Triangular Automaton Rule 170
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015.
Alexey Kovba, Illustration for n = 0..5
MohammadReza Saadat and Benedek Nagy, Copy Machines - Self-reproduction with 2 States on Archimedean Tilings, Journal of Cellular Automata, vol. 17, pp. 221-249, 2023.
Index to sequences in the OEIS related to cellular automata

Cf. A160239 (square tiling analog), A247640, A247666 (hexagonal tiling analogs).
Cf. A000244, A011782, A102900, A382971.
Pattern replicating ETA rules: A383369 (rule 90).

a(0) = 1. Conjecture: a(2*t+1) = 3*a(t).

Conjectures: a(8*t+6) = 3*a(4*t+2) + 4*a(2*t), a(8*t+2) = 3*a(4*t) + 4*a(2*t), a(4*t) = a(2*t). These conjectured formulas together give recurrent relations for a(n) for any n. Also, obviously a(2*n) = A247666(n). - Andrey Zabolotskiy, Aug 04 2016

A246039 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+y)+1.

Original entry on oeis.org

1, 7, 7, 29, 7, 49, 29, 103, 7, 49, 49, 203, 29, 203, 103, 373, 7, 49, 49, 203, 49, 343, 203, 721, 29, 203, 203, 841, 103, 721, 373, 1407, 7, 49, 49, 203, 49, 343, 203, 721, 49, 343, 343, 1421, 203, 1421, 721, 2611, 29, 203, 203, 841, 203, 1421, 841, 2987, 103, 721, 721, 2987, 373, 2611, 1407, 5277

Offset: 0

N. J. A. Sloane, Aug 21 2014

nonn

This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.
This is the odd-rule cellular automaton defined by OddRule 575 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
Run Length Transform of A246038.
The Run Length Transform of a sequence {S(n), n>=0} is defined to be the sequence {T(n), n>=0} given by T(n) = Product_i S(i), where i runs through the lengths of runs of 1's in the binary expansion of n. E.g. 19 is 10011 in binary, which has two runs of 1's, of lengths 1 and 2. So T(19) = S(1)*S(2). T(0)=1 (the empty product).

			Here is the neighborhood:
[X, X, X]
[0, X, 0]
[X, X, X]
which contains a(1) = 7 ON cells.

Alois P. Heinz, Table of n, a(n) for n = 0..8192
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796, 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249, 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
Index entries for sequences related to cellular automata

Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035. A246037.

Cf. A246038.

C:=f->subs({x=1, y=1}, f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x, y).
OddCA:=proc(f, M) global C; local n, a, i, f2, p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1;
for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i], i=1..nops(a))]);
end;
f:=(1/x+1+x)*(1/y+y)+1 mod 2;
OddCA(f, 70);

(* f = A246038 *) f[0]=1; f[1]=7; f[2]=29; f[3]=103; f[4]=373; f[n_] := f[n] = 8 f[n-4] + 8 f[n-3] + 3 f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* Jean-François Alcover, Jul 12 2017 *)

A246314 Number of odd terms in f^n, where f = 1/x^2+1/x+1+x+x^2+1/y^2+1/y+y+y^2.

Original entry on oeis.org

1, 9, 9, 37, 9, 65, 37, 157, 9, 81, 65, 237, 37, 293, 157, 713, 9, 81, 81, 333, 65, 473, 237, 1077, 37, 333, 293, 1129, 157, 1285, 713, 2737, 9, 81, 81, 333, 81, 585, 333, 1413, 65, 585, 473, 1733, 237, 1933, 1077, 4337, 37, 333, 333, 1369, 293, 2125, 1129, 4969, 157, 1413, 1285, 5041, 713, 5561, 2737, 11421, 9, 81

Offset: 0

N. J. A. Sloane, Aug 26 2014

nonn

This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f (a cross containing 9 cells), and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.

			Here is the neighborhood:
[0, 0, X, 0, 0]
[0, 0, X, 0, 0]
[X, X, X, X, X]
[0, 0, X, 0, 0]
[0, 0, X, 0, 0]
which contains a(1) = 9 ON cells.
The second and third generations are:
[0, 0, 0, 0, X, 0, 0, 0, 0]
[0, 0, 0, 0, 0, 0, 0, 0, 0]
[0, 0, 0, 0, X, 0, 0, 0, 0]
[0, 0, 0, 0, 0, 0, 0, 0, 0]
[X, 0, X, 0, X, 0, X, 0, X]  (again with 9 ON cells)
[0, 0, 0, 0, 0, 0, 0, 0, 0]
[0, 0, 0, 0, X, 0, 0, 0, 0]
[0, 0, 0, 0, 0, 0, 0, 0, 0]
[0, 0, 0, 0, X, 0, 0, 0, 0]
[0, 0, 0, 0, 0, 0, X, 0, 0, 0, 0, 0, 0]
[0, 0, 0, 0, 0, 0, X, 0, 0, 0, 0, 0, 0]
[0, 0, 0, 0, X, X, 0, X, X, 0, 0, 0, 0]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[0, 0, X, 0, 0, X, X, X, 0, 0, X, 0, 0]
[0, 0, X, 0, X, 0, 0, 0, X, 0, X, 0, 0]
[X, X, 0, 0, X, 0, X, 0, X, 0, 0, X, X] (with 37 ON cells)
[0, 0, X, 0, X, 0, 0, 0, X, 0, X, 0, 0]
[0, 0, X, 0, 0, X, X, X, 0, 0, X, 0, 0]
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[0, 0, 0, 0, X, X, 0, X, X, 0, 0, 0, 0]
[0, 0, 0, 0, 0, 0, X, 0, 0, 0, 0, 0, 0]
[0, 0, 0, 0, 0, 0, X, 0, 0, 0, 0, 0, 0]
The terms can be arranged into blocks of sizes 1,1,2,4,8,16,32,...:
1,
9,
9, 37,
9, 65, 37, 157,
9, 81, 65, 237, 37, 293, 157, 713,
9, 81, 81, 333, 65, 473, 237, 1077, 37, 333, 293, 1129, 157, 1285, 713, 2737,
9, 81, 81, 333, 81, 585, 333, 1413, 65, 585, 473, 1733, 237, 1933, 1077, 4337, 37, 333, 333, 1369, 293, 2125, 1129, 4969, 157, 1413, 1285, 5041, 713, 5561, 2737, 11421, ...
The final terms in the rows are A246315.

Alois P. Heinz, Table of n, a(n) for n = 0..512

Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035, A246037.

Cf. A246315, A247647, A247648, A247649.

C:=f->subs({x=1, y=1}, f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x, y).
OddCA:=proc(f, M) global C; local n, a, i, f2, p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1;
for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i], i=1..nops(a))]);
end;
f:=1/x^2+1/x+1+x+x^2+1/y^2+1/y+y+y^2;
OddCA(f, 70);

c[f_] := f /. {x -> 1, y -> 1};
OddCA[f_, M_] := Module[{a = {}, f2, p = 1}, f2 = PolynomialMod[f, 2]; Do[ AppendTo[a, c[p]]; Print[a]; p = PolynomialMod[p f2, 2], {n, 0, M}]; a];
f = 1/x^2 + 1/x + 1 + x + x^2 + 1/y^2 + 1/y + y + y^2;
OddCA[f, 70] (* Jean-François Alcover, May 24 2020, after Maple *)

The values of a(n) for n in A247647 (or A247648) determine all the values, as follows. Parse the binary expansion of n into terms from A247647 separated by at least two zeros: m_1 0...0 m_2 0...0 m_3 ... m_r 0...0. Ignore any number (one or more) of trailing zeros. Then a(n) = a(m_1)*a(m_2)*...*a(m_r). For example, n = 37_10 = 100101_2 is parsed into 1.00.101, and so a(37) = a(1)*a(5) = 9*65 = 585. This is a generalization of the Run Length Transform.

A253069 Number of odd terms in f^n, where f = 1/x+1+x+x/y+y/x+x*y.

Original entry on oeis.org

1, 6, 6, 22, 6, 36, 22, 82, 6, 36, 36, 132, 22, 132, 82, 302, 6, 36, 36, 132, 36, 216, 132, 492, 22, 132, 132, 484, 82, 492, 302, 1106, 6, 36, 36, 132, 36, 216, 132, 492, 36, 216, 216, 792, 132, 792, 492, 1812, 22, 132, 132, 484, 132, 792, 484, 1804, 82, 492, 492, 1804, 302, 1812, 1106, 4066

Offset: 0

N. J. A. Sloane, Jan 29 2015

nonn

This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.

This is the odd-rule cellular automaton defined by OddRule 175 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).

			Here is the neighborhood f:
[X, 0, X]
[X, X, X]
[0, 0, X]
which contains a(1) = 6 ON cells.

Alois P. Heinz, Table of n, a(n) for n = 0..8191
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796, 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249, 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
Index entries for sequences related to cellular automata

Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035, A253064, A253065, A253066.

Cf. A253070.

C:=f->subs({x=1, y=1}, f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x, y).
OddCA:=proc(f, M) global C; local n, a, i, f2, p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1;
for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i], i=1..nops(a))]);
end;
f:=1/x+1+x+x/y+y/x+x*y;
OddCA(f, 130);

(* f = A253070 *) f[0]=1; f[1]=6; f[2]=22; f[3]=82; f[4]=302; f[5]=1106;f[6]=4066; f[n_] := f[n] = 8 f[n-4] + 8 f[n-3] + 3 f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* Jean-François Alcover, Jul 12 2017 *)

This is the Run Length Transform of A253070.

A253071 Number of odd terms in f^n, where f = 1/(xy)+1/x+1/y+y+x/y+x+xy.

Original entry on oeis.org

1, 7, 7, 21, 7, 49, 21, 95, 7, 49, 49, 147, 21, 147, 95, 333, 7, 49, 49, 147, 49, 343, 147, 665, 21, 147, 147, 441, 95, 665, 333, 1319, 7, 49, 49, 147, 49, 343, 147, 665, 49, 343, 343, 1029, 147, 1029, 665, 2331, 21, 147, 147, 441, 147, 1029, 441, 1995, 95, 665, 665, 1995, 333, 2331, 1319, 4837

Offset: 0

N. J. A. Sloane and Doron Zeilberger, Feb 23 2015

nonn

This is the number of ON cells in a certain 2-D CA in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there was an odd number of ON cells in the neighborhood at the previous generation.

This is the odd-rule cellular automaton defined by OddRule 357 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).

			Here is the neighborhood f:
[0, X, X]
[X, 0, X]
[X, X, X]
which contains a(1) = 7 ON cells.

Alois P. Heinz, Table of n, a(n) for n = 0..8191
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796, 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249, 2015.
N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
Index entries for sequences related to cellular automata

Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035, A253064, A253065, A253066, A252069.

Cf. A253072.

C:=f->subs({x=1, y=1}, f);
# Find number of ON cells in CA for generations 0 thru M defined by rule
# that cell is ON iff number of ON cells in nbd at time n-1 was odd
# where nbd is defined by a polynomial or Laurent series f(x, y).
OddCA:=proc(f, M) global C; local n, a, i, f2, p;
f2:=simplify(expand(f)) mod 2;
a:=[]; p:=1;
for n from 0 to M do a:=[op(a), C(p)]; p:=expand(p*f2) mod 2; od:
lprint([seq(a[i], i=1..nops(a))]);
end;
f:=1/(x*y)+1/x+1/y+y+x/y+x+x*y;
OddCA(f, 130);

(* f = A253072 *) f[0]=1; f[1]=7; f[2]=21; f[3]=95; f[4]=333; f[5]=1319; f[n_] := f[n] = -8 f[n-5] + 44 f[n-4] - 24 f[n-3] - 5 f[n-2] + 6 f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* Jean-François Alcover, Jul 12 2017 *)

This is the Run Length Transform of A253072.

A255289 Number of 1's in expansion of F^n mod 3, where F = 1/(xy)+2/y+x/y+2/x+2x+y/x+2y+xy.

Original entry on oeis.org

1, 4, 12, 4, 32, 48, 12, 84, 117, 4, 32, 84, 32, 256, 300, 48, 336, 324, 12, 84, 225, 84, 672, 792, 117, 852, 876, 4, 32, 84, 32, 256, 336, 84, 672, 852, 32, 256, 672, 256, 2048, 2316, 300, 2352, 2448, 48, 336, 900, 336

Offset: 0

N. J. A. Sloane, Feb 21 2015

nonn

A255289 and A255290 together are a second mod 3 analog of A160239.

			The pairs [no. of 1's, no. of 2's] are [1, 0], [4, 4], [12, 9], [4, 4], [32, 32], [48, 36], [12, 9], [84, 84], [117, 96], [4, 4], [32, 32], [84, 84], [32, 32], [256, 256], [300, 288], [48, 36], [336, 336], [324, 420], [12, 9], [84, 84], [225, 216], [84, 84], [672, 672], [792, 744], [117, 96], [852, 852], [876, 1197], ...

Cf. A160239, A255287-A255294.

# C3 Counts 1's and 2's
C3 := proc(f) local c,ix,iy,f2,i,t1,t2,n1,n2;
f2:=expand(f) mod 3; n1:=0; n2:=0;
if whattype(f2) = `+` then
t1:=nops(f2);
for i from 1 to t1 do t2:=op(i, f2); ix:=degree(t2, x); iy:=degree(t2, y);
c:=coeff(coeff(t2,x,ix),y,iy);
if (c mod 3) = 1 then n1:=n1+1; else n2:=n2+1; fi; od: RETURN([n1,n2]);
else ix:=degree(f2, x); iy:=degree(f2, y);
c:=coeff(coeff(f2,x,ix),y,iy);
if (c mod 3) = 1 then n1:=n1+1; else n2:=n2+1; fi; RETURN([n1,n2]);
fi;
end;
F2:=1/(x*y)+2/y+x/y+2/x+2*x+y/x+2*y+x*y mod 3;
g:=(F,n)->expand(F^n) mod 3;
[seq(C3(g(F2,n))[1],n=0..60)];

A255290 Number of 2's in expansion of F^n mod 3, where F = 1/(xy)+2/y+x/y+2/x+2x+y/x+2y+xy.

Original entry on oeis.org

0, 4, 9, 4, 32, 36, 9, 84, 96, 4, 32, 84, 32, 256, 288, 36, 336, 420, 9, 84, 216, 84, 672, 744, 96, 852, 1197, 4, 32, 84, 32, 256, 336, 84, 672, 852, 32, 256, 672, 256, 2048, 2304, 288, 2352, 2544, 36, 336, 864, 336, 2688

Offset: 0

N. J. A. Sloane, Feb 21 2015

nonn

A255289 and A255290 together are a second mod 3 analog of A160239.

			The pairs [no. of 1's, no. of 2's] are [1, 0], [4, 4], [12, 9], [4, 4], [32, 32], [48, 36], [12, 9], [84, 84], [117, 96], [4, 4], [32, 32], [84, 84], [32, 32], [256, 256], [300, 288], [48, 36], [336, 336], [324, 420], [12, 9], [84, 84], [225, 216], [84, 84], [672, 672], [792, 744], [117, 96], [852, 852], [876, 1197], ...

Cf. A160239, A255287-A255294.

# C3 Counts 1's and 2's
C3 := proc(f) local c,ix,iy,f2,i,t1,t2,n1,n2;
f2:=expand(f) mod 3; n1:=0; n2:=0;
if whattype(f2) = `+` then
t1:=nops(f2);
for i from 1 to t1 do t2:=op(i, f2); ix:=degree(t2, x); iy:=degree(t2, y);
c:=coeff(coeff(t2,x,ix),y,iy);
if (c mod 3) = 1 then n1:=n1+1; else n2:=n2+1; fi; od: RETURN([n1,n2]);
else ix:=degree(f2, x); iy:=degree(f2, y);
c:=coeff(coeff(f2,x,ix),y,iy);
if (c mod 3) = 1 then n1:=n1+1; else n2:=n2+1; fi; RETURN([n1,n2]);
fi;
end;
F2:=1/(x*y)+2/y+x/y+2/x+2*x+y/x+2*y+x*y mod 3;
g:=(F,n)->expand(F^n) mod 3;
[seq(C3(g(F2,n))[2],n=0..60)];

A164032 Number of "ON" cells in a certain 2-dimensional cellular automaton.

Original entry on oeis.org

1, 9, 4, 36, 4, 36, 16, 144, 4, 36, 16, 144, 16, 144, 64, 576, 4, 36, 16, 144, 16, 144, 64, 576, 16, 144, 64, 576, 64, 576, 256, 2304, 4, 36, 16, 144, 16, 144, 64, 576, 16, 144, 64, 576, 64, 576, 256, 2304, 16, 144, 64, 576, 64, 576, 256, 2304, 64, 576, 256, 2304, 256

Offset: 1

John W. Layman, Aug 08 2009

nonn

This automaton starts with one ON cell and evolves according to the rule that a cell is ON in a given generation if and only if the number of ON cells, among the cell itself and its eight nearest neighbors, was exactly one in the preceding generation.

			Can be arranged into blocks of length 2^k:
1,
9,
4, 36,
4, 36, 16, 144,
4, 36, 16, 144, 16, 144, 64, 576,
4, 36, 16, 144, 16, 144, 64, 576, 16, 144, 64, 576, 64, 576, 256, 2304,
4, 36, 16, 144, 16, 144, 64, 576, 16, 144, 64, 576, 64, 576, 256, 2304, 16, 144, 64, 576, 64, 576, 256, 2304, 64, 576, 256, 2304, 256, ...
...

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.], which is also available at arXiv:1004.3036v2
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A000120, A048883, A079315, A122108, A160239, A002063 (last entry in each block)

wt[i_] := DigitCount[i, 2, 1];
a[n_] := If[OddQ[n], 1, 9] 4^wt[Floor[(n-1)/2]];
Array[a, 61] (* Jean-François Alcover, Oct 08 2018, after N. J. A. Sloane *)

a(n) = 4^hammingweight((n-1)\2) * if(n%2, 1, 9); \\ Michel Marcus, Oct 08 2018

It appears that this is the self-generating sequence defined by the following process: start with s={1,9} and repeatedly extend by concatenating s with 4*s, thus obtaining {1,9} -> {1,9,4,36} -> {1,9,4,36,4,36,16,144},... , etc.
Also, it appears that if n=2^k+j, with n>2 and 1<=j<=2^k, then a(n)=4a(j), with a(1)=1, a(2)=9.
From N. J. A. Sloane, Jul 21 2014: (Start)
Both of these assertions are not difficult to prove. At generation G = 2^k (k>=1) the ON cells are bounded by a box of edge 2G-1, and in that box there are (G/2)^2 3X3 blocks each containing 9 ON cells (separated by rows of OFF cells of width 1), so a total of a(2^k) = 9*2^(2k-2) ON cells (cf. A002063).
This box is full (more precisely, every cell in it has more than one ON neighbor), and at generation G+1 we have just 4 ON cells which are now at the corners of a box of edge 2G+1. Until the next power of 2 there is no interaction between the configurations that grow at the four corners, and so a(2^k+j) = 4a(j), as conjectured.
In fact this implies an explicit formula for a(n):
a(n) = c*4^wt(floor((n-1)/2)),
where c=1 if n is odd, c=9 if n is even, and wt(i) = A000120(i) is the binary weight function. For example, if n=20, [(n-1)/2]=9 which has weight 2, so a(20) = 9*4^2 = 144. (End)

A254731 Number of ON cells in the even-rule cellular automaton after n steps with the Moore neighborhood (8 neighbors), with minimal nontrivial symmetric initial state (0,0), (0,1), (1,0), and (1,1) ON.

Original entry on oeis.org

4, 8, 24, 20, 32, 68, 48, 72, 116, 88, 104, 140, 188, 160, 284, 272, 268, 320, 372, 352, 496, 488, 524, 608, 556, 628, 692, 820, 764, 808, 864, 976, 1024, 920, 1032, 1228, 1188, 1256, 1408, 1496, 1488, 1564, 1584, 1712, 1752, 1708, 1888, 2148, 2040, 2100, 2308, 2392, 2544, 2480, 2760, 2752, 2764, 3064, 3020, 2976, 3516, 3440, 3560, 3580, 3804, 3816, 3916, 4236, 4492, 4340, 4516, 4512, 4984, 4764, 5004, 4880, 5116, 5716, 5540, 5560, 5564, 5840, 6200, 6368, 6280, 6668, 6880, 6908, 6960, 7600, 7388, 7396, 8028, 7832, 8332, 8152, 8268, 8928, 8708, 9144

Offset: 0

Kellen Myers, Feb 06 2015

nonn

The rule turns a cell to ON at step n if an even, nonzero number of its eight neighbors were ON in the previous. For example, at n=2 the cell (0,0) is ON because the two neighbors (-1,0) and (0,-1) and no others were ON at the previous step.

It appears that whenever n is divisible by 3, there is a visible disjoint 2x2 square leading the automaton in each cardinal direction.

			For n=3, the configuration includes the initial four ON cells plus four other 2 X 2 squares in each cardinal direction.

Index entries for sequences related to cellular automata

Cf. A160239.

m = 100; n = 2 m + 1;
A = Table[0, {p, 1, m}, {q, 1, n}, {z, 1, n}];
A[[1, m, m + 1]] = 1;
A[[1, m, m]] = 1;
A[[1, m + 1, m + 1]] = 1;
A[[1, m + 1, m]] = 1;
For[i = 2, i <= m, i++,
For[x = 2, x <= n - 1, x++,
  For[y = 2, y <= n - 1, y++,
   sum = A[[i - 1, x - 1, y - 1]] +
     A[[i - 1, x, y - 1]] +
     A[[i - 1, x + 1, y - 1]] +
     A[[i - 1, x - 1, y]] +
     A[[i - 1, x + 1, y]] +
     A[[i - 1, x - 1, y + 1]] +
     A[[i - 1, x, y + 1]] +
     A[[i - 1, x + 1, y + 1]];
   A[[i, x, y]] = If[sum > 0, 1 - Mod[sum, 2], 0];
   ]
  ]
];
Table[Plus @@ Plus @@ A[[i, All, All]], {i, 1, m}]
(* Kellen Myers, Feb 07 2015 *)

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