A072272 -id:A072272 - cp's OEIS frontend

A246037 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+y).

1, 6, 6, 20, 6, 36, 20, 88, 6, 36, 36, 120, 20, 120, 88, 336, 6, 36, 36, 120, 36, 216, 120, 528, 20, 120, 120, 400, 88, 528, 336, 1376, 6, 36, 36, 120, 36, 216, 120, 528, 36, 216, 216, 720, 120, 720, 528, 2016, 20, 120, 120, 400, 120, 720, 400, 1760, 88, 528, 528, 1760, 336, 2016, 1376, 5440

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 077 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).
Run Length Transform of A246036.
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, 0, 0]
[X, X, X]
which contains a(1) = 6 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 [math.CO], 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 [math.CO], 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 [math.CO], 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.

Cf. A246036.

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);
OddCA(f, 70);

(* f = A246036 *) f[0] = 1; f[n_] := (4^(n+1)-(-2)^n)/3; Table[Times @@ (f[Length[#]]&) /@ Select[s = Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* Jean-François Alcover, Jul 12 2017 *)

A269522 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 622", based on the 5-celled von Neumann neighborhood.

Original entry on oeis.org

1, 5, 9, 21, 25, 37, 57, 69, 89, 101, 121, 133, 169, 205, 257, 309, 361, 333, 377, 381, 385, 461, 465, 509, 601, 653, 697, 781, 889, 845, 1097, 1077, 1353, 1205, 1305, 1349, 1345, 1357, 1321, 1437, 1553, 1589, 1601, 1709, 1729, 1813, 1937, 2069, 2361, 2181

Offset: 0

Robert Price, Feb 28 2016

Initialized with a single black (ON) cell at stage zero.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Robert Price, Table of n, a(n) for n = 0..300
Robert Price, Diagrams of first 20 stages
N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168, 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata

Cf. A007483, A072272, A170878, A253908.

rule=622; stages=300; ca=CellularAutomaton[ {rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}}, {{{1}},0}, stages]; (* Start with single black cell *) Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)

A253090 Number of ON cells at generation n of 9-celled totalistic CA defined by Rule 614.

Original entry on oeis.org

1, 9, 8, 32, 16, 88, 16, 104, 16, 136, 72, 312, 56, 360, 76, 468, 76, 444, 156, 708, 152, 928, 164, 996, 156, 924, 292, 1316, 256, 1520, 348, 1676, 304, 1768, 400, 1928, 408, 2040, 524, 2628, 516, 2684, 688, 3272, 744, 3872, 684, 3692, 756, 3724, 840, 4080

Offset: 0

N. J. A. Sloane, Feb 12 2015

nonn

A cell turns ON at generation n iff it was OFF at generation n-1 and exactly 1 or 3 of its 8 neighbors were ON, or if it was ON and exactly 0, 2 or 4 of its 8 neighbors were ON.

Note that although A072272 is an "odd-rule" CA, this one is not.

N. J. A. Sloane, Illustration of generations 0-15
N. J. A. Sloane, Illustration of generation 31
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. A072272 (5-neighbor analog).

Map[Function[Apply[Plus, Flatten[#1]]],
CellularAutomaton[{614, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, 66]]
ArrayPlot /@ CellularAutomaton[{614, {2, {{2, 2, 2}, {2, 1, 2}, {2, 2, 2}}}, {1, 1}}, {{{1}}, 0}, 15]

It would be nice to have a recurrence.

A138277 Total number of active nodes of the Rule 150 cellular automaton on an infinite Bethe lattice with coordination number 4 (with a single 1 as initial condition).

Original entry on oeis.org

1, 5, 13, 49, 109, 473, 1081, 4037, 8749, 37913, 88465, 325021, 717337, 3108461, 7095613, 26490289, 57395629, 248714393, 580333585, 2132141341, 4707150193, 20397650837, 46548642709, 173816036825, 376630110937, 1632063814061, 3808148899477, 13991111158153

Offset: 0

Jens Christian Claussen (claussen(AT)theo-physik.uni-kiel.de), Mar 11 2008

nonn

See A138276 for the corresponding sequence for a Bethe lattice with coordination number 3.
See A001045 for the corresponding sequence on a 1D lattice (equivalent to a k=2 Bethe lattice); this is based on the Jacobsthal sequence A001045.
See A072272 for the corresponding sequence on a 2D lattice (based on A007483).
Related to Cellular Automata.

			Let x_0 be the state (0 or 1) of the focal node and x_i the state of every node that is i steps away from the focal node. In time step n=0, all x_i=0 except x_0=1 (start with a single seed). In the next step, x_1=1 as they have 1 neighbor being 1. For n=2, the x_1 nodes have 1 neighbor being 1 (x_0) and themselves being 1; the sum being 2, modulo 2, resulting in x_1=0.
The focal node and outmost nodes x_n are always 1.
Thus one has the patterns
x_0, x_1, x_2, ...
1
1 1
1 0 1
1 0 1 1
1 0 0 0 1
1 1 0 1 1 1
1 0 0 0 1 0 1
1 1 0 1 1 0 1 1
1 0 0 0 0 0 0 0 1
(N.B.: This is equivalent to the right half plane of Rule 150 in 1D.)
The nodes have the multiplicities 1,4,12,36,108,324,972,...
The sequence then is obtained by
a(n)= x_0(n) + 4*(x_1(n) + sum_(i=2...n) x_i(n) * 3^(i-1)).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Jens Christian Claussen, Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration [broken link]
Jens Christian Claussen, Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration, arXiv:math.CO/0410429.
Jan Nagler and Jens Christian Claussen (2005), 1/f^alpha spectra in elementary cellular automata and fractal signals, Phys. Rev. E 71 (2005), 067103

Cf. A138276, A072272, A007483, A071053, A001045.

nmax = 30;
states = CellularAutomaton[150, {{1}, 0}, nmax];
T[n_, i_] := states[[n+1, nmax+i+1]];
a[n_] := T[n, 0] + 4(T[n, 1]+Sum[3^(i-1) T[n, i], {i, 2, n}]);
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Aug 20 2018 *)

The total number of nodes in state 1 after n iterations (starting with a single 1) of the Rule 150 cellular automaton on an infinite Bethe lattice with coordination number 4. Rule 150 sums the values of the focal node and its k neighbors, then applies modulo 2.

a(9)-a(27) from Alois P. Heinz, Jun 28 2015

A165345 Number of ON cells after n generations of the 2D cellular automaton described in the comments.

Original entry on oeis.org

1, 5, 9, 25, 29, 41, 53, 105, 113, 129, 141, 193, 205, 241, 285, 433, 453, 481, 497, 553, 569, 609, 653, 801, 829, 881, 917, 1073, 1109, 1217, 1349, 1793, 1845, 1905, 1933, 2001, 2029, 2081, 2129, 2281, 2313, 2369, 2409, 2569, 2609, 2721, 2853, 3297, 3357

Offset: 1

John W. Layman, Sep 15 2009, Sep 16 2009

nonn

The cells are the squares of the standard square grid. All cells are initially OFF and one cell is turned ON at generation 1. At subsequent generations a cell is ON if and only if (1) it was ON, or (2) exactly one of the four nearest side neighbors was ON, or (3) exactly three of the four nearest corner neighbors were ON, in the previous generation

The equivalent Mathematica automaton is obtained with neighborhood weights {{10,2,10},{2,1,2},{10,2,10}}, rule number 755364134566574, and initial configuration {{1}} (see code).

Graphs of the automaton for generations 1-20. [From _John W. Layman_, Sep 15 2009]

Cf. A048883, A072272, A122108, A160410, A164982.

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 = {{10, 2, 10}, {2, 1, 2}, {10, 2, 10}}; rule=755364134566574; init = {{1}}; Show[GraphicsArray[ Map[RasterGraphics, CellularAutomaton[{rule, {2, wt}, {1, 1}}, {init, 0}, 9, -10]]]]; ca = CellularAutomaton[{rule, {2, wt}, {1, 1}}, {init, 0}, 99, -100]; a = Table[Total[ca[[i]], 2], {i, 1, 100}]

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.

A255291 Number of 1's in expansion of F^n mod 3, where F = 1/x+1+x+1/y+y.

Original entry on oeis.org

1, 5, 4, 5, 25, 12, 4, 20, 69, 5, 25, 20, 25, 125, 52, 12, 60, 281, 4, 20, 97, 20, 100, 353, 69, 345, 448, 5, 25, 20, 25, 125, 60, 20, 100, 345, 25, 125, 100, 125, 625, 252, 52, 260, 1341, 12, 60, 381, 60, 300, 1413, 281, 1405

Offset: 0

N. J. A. Sloane, Feb 21 2015

nonn

A255291 and A255292 together are a mod 3 analog of A072272.

			The pairs [no. of 1's, no. of 2's] are [1, 0], [5, 0], [4, 9], [5, 0], [25, 0], [12, 37], [4, 9], [20, 45], [69, 44], [5, 0], [25, 0], [20, 45], [25, 0], [125, 0], [52, 177], [12, 37], [60, 185], [281, 156], [4, 9], [20, 45], [97, 72], [20, 45], [100, 225], [353, 228], [69, 44], [345, 220], [448, 573], [5, 0], [25, 0], [20, 45], ...

Cf. A072272, A255288-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;
F3:=1/x+1+x+1/y+y mod 3;
g:=(F,n)->expand(F^n) mod 3;
[seq(C3(g(F3,n))[1],n=0..60)];

cp's OEIS Frontend

A246037 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+y).

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A269522 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 622", based on the 5-celled von Neumann neighborhood.

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A253090 Number of ON cells at generation n of 9-celled totalistic CA defined by Rule 614.

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A138277 Total number of active nodes of the Rule 150 cellular automaton on an infinite Bethe lattice with coordination number 4 (with a single 1 as initial condition).

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A165345 Number of ON cells after n generations of the 2D cellular automaton described in the comments.

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

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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.

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

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