A071053 -id:A071053 - cp's OEIS frontend

A118110 State of one-dimensional cellular automaton 'sigma' (Rule 150): 000,001,010,011,100,101,110,111 -> 0,1,1,0,1,0,0,1 at generation n, when started with a single ON cell, regarded as a binary number.

1, 111, 10101, 1101011, 100010001, 11101110111, 1010001000101, 110110111011011, 10000000100000001, 1110000011100000111, 101010001010100010101, 11010110110101101101011, 1000100000001000000010001

Offset: 0

Eric W. Weisstein, Apr 13 2006 and N. J. A. Sloane, Jul 28 2014, combined into one entry by N. J. A. Sloane, Oct 20 2015

nonn

See A038184 for decimal equivalents.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.

Eric Weisstein's World of Mathematics, Rule 150
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Index to Elementary Cellular Automata
Index entries for sequences related to cellular automata

This sequence, A038184 and A071036 are equivalent descriptions of the Rule 150 automaton.

See A071053 for number of ON cells.

rule = 150; rows = 20; Table[FromDigits[Table[Take[CellularAutomaton[rule, {{1}, 0}, rows-1, {All, All}][[k]], {rows-k+1, rows+k-1}], {k, 1, rows}][[k]]], {k, 1, rows}] (* Robert Price, Feb 21 2016 *)

A246028 a(n) = Product_{i in row n of A245562} Fibonacci(i+1).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 2, 3, 5, 1, 1, 1, 2, 1, 1, 2, 3, 2, 2, 2, 4, 3, 3, 5, 8, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 2, 3, 5, 2, 2, 2, 4, 2, 2, 4, 6, 3, 3, 3, 6, 5, 5, 8, 13, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 2, 3, 5, 1, 1, 1, 2, 1, 1, 2, 3, 2, 2, 2, 4, 3, 3, 5, 8, 2, 2, 2, 4, 2

Offset: 0

N. J. A. Sloane, Aug 15 2014; revised Sep 05 2014

This is the Run Length Transform of S(n) = Fibonacci(n+1).

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).

Alois P. Heinz, Table of n, a(n) for n = 0..8191
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
Chai Wah Wu, Sums of products of binomial coefficients mod 2 and run length transforms of sequences, arXiv:1610.06166 [math.CO], 2016.
Index entries for sequences related to cellular automata

Cf. A245562, A000045, A001045, A071053, A245565, A245564.

with(combinat); ans:=[];
for n from 0 to 100 do lis:=[]; t1:=convert(n,base,2); L1:=nops(t1); out1:=1; c:=0;
for i from 1 to L1 do
if out1 = 1 and t1[i] = 1 then out1:=0; c:=c+1;
elif out1 = 0 and t1[i] = 1 then c:=c+1;
elif out1 = 1 and t1[i] = 0 then c:=c;
elif out1 = 0 and t1[i] = 0 then lis:=[c,op(lis)]; out1:=1; c:=0;
fi;
if i = L1 and c>0 then lis:=[c,op(lis)]; fi;
od:
a:=mul(fibonacci(i+1), i in lis);
ans:=[op(ans),a];
od:
ans;

a[n_] := Sum[Mod[Binomial[n-k, 2k] Binomial[n, k], 2], {k, 0, n}];
a /@ Range[0, 100] (* Jean-François Alcover, Feb 28 2020, after Chai Wah Wu *)

a(n)=my(s=1,k); while(n, n>>=valuation(n,2); k=valuation(n+1,2); if(k>1, s*=fibonacci(k+1)); n>>=k); s \\ Charles R Greathouse IV, Oct 21 2016

a(n)=sum(k=0,n, !bitand(n-3*k,2*k) && !bitand(n-k,k)) \\ Charles R Greathouse IV, Oct 21 2016

def A246028(n): return sum(int(not (~(n-k) & 2*k) | (~n & k)) for k in range(n+1)) # Chai Wah Wu, Sep 27 2021

a(n) = Sum_{k=0..n} ((binomial(n-k,2k)*binomial(n,k)) mod 2). - Chai Wah Wu, Oct 19 2016

A246685 Run Length Transform of sequence 1, 3, 5, 17, 257, 65537, ... (1 followed by Fermat numbers).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 1, 3, 3, 3, 5, 17, 1, 1, 1, 3, 1, 1, 3, 5, 3, 3, 3, 9, 5, 5, 17, 257, 1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 1, 3, 3, 3, 5, 17, 3, 3, 3, 9, 3, 3, 9, 15, 5, 5, 5, 15, 17, 17, 257, 65537, 1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 1, 3, 3, 3, 5, 17, 1, 1, 1, 3, 1, 1, 3, 5, 3, 3, 3, 9, 5, 5, 17, 257

Offset: 0

Antti Karttunen, Sep 22 2014

nonn

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).

This sequence is obtained by applying Run Length Transform to sequence b = 1, 3, 5, 17, 257, 65537, ... (1 followed by Fermat numbers, with b(1) = 1, b(2) = 3, b(3) = 5, ..., b(n) = 2^(2^(n-2)) + 1 for n >= 2).

			115 is '1110011' in binary. The run lengths of 1-runs are 2 and 3, thus we multiply the second and the third elements of the sequence 1, 3, 5, 17, 257, 65537, ... to get a(115) = 3*5 = 15.

Antti Karttunen, Table of n, a(n) for n = 0..1024

Cf. A003714 (gives the positions of ones).
Cf. A000215.
A001316 is obtained when the same transformation is applied to A000079, the powers of two. Cf. also A001317.
Run Length Transforms of other sequences: A071053, A227349, A246588, A246595, A246596, A246660, A246661, A246674, A247282.

f[n_] := Switch[n, 0|1, 1, , 2^(2^(n-2))+1]; Table[Times @@ (f[Length[#]] &) /@ Select[s = Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 95}] (* _Jean-François Alcover, Jul 11 2017 *)

# use RLT function from A278159
def A246685(n): return RLT(n,lambda m: 1 if m <= 1 else 2**(2**(m-2))+1) # Chai Wah Wu, Feb 04 2022

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

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.

A071052 Number of 0's in n-th row of triangle in A071036 (cellular automaton "Rule 150").

Original entry on oeis.org

0, 0, 2, 2, 6, 2, 8, 4, 14, 10, 12, 8, 20, 12, 18, 10, 30, 26, 28, 24, 32, 16, 30, 14, 44, 36, 38, 30, 46, 26, 40, 20, 62, 58, 60, 56, 64, 48, 62, 46, 72, 56, 58, 42, 74, 46, 60, 32, 92, 84, 86, 78, 90, 62, 84, 56, 102, 82, 84, 64, 100, 60, 82, 42, 126, 122, 124

Offset: 0

Hans Havermann, May 26 2002

nonn

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; Chapter 3.

Index entries for sequences related to cellular automata

Cf. A071053 (number of 1's).

a(2n) = a(n) + 2n (conjectured). - Ralf Stephan, Mar 03 2004

A118911 Row sums of triangle in A128973.

Original entry on oeis.org

1, 1, 3, 3, 3, 3, 5, 5, 3, 3, 9, 9, 5, 5, 11, 11, 3, 3, 9, 9, 9, 9, 15, 15, 5, 5, 15, 15, 11, 11, 21, 21, 3, 3, 9, 9, 9, 9, 15, 15, 9, 9, 27, 27, 15, 15, 33, 33, 5, 5, 15, 15, 15, 15, 25, 25, 11, 11, 33, 33, 21, 21, 43, 43, 3, 3, 9, 9, 9, 9, 15, 15, 9, 9, 27, 27, 15, 15, 33, 33, 9, 9

Offset: 0

Philippe Deléham, May 02 2007, Jan 06 2008

nonn

Integers in A071053 repeated.

Cf. A071053, A128973.

a(2*n) = a(2*n+1).
a(n) = Sum_{k=0..n} A128973(n,k).
a(2^k+m) = 3*a(m) for 0 <= m <= 2^(k-1) - 1, k >= 1.
a(4*n) = a(2*n) = A071053(n).

a(48) and following terms corrected by Georg Fischer, Jul 04 2023

cp's OEIS Frontend

A118110 State of one-dimensional cellular automaton 'sigma' (Rule 150): 000,001,010,011,100,101,110,111 -> 0,1,1,0,1,0,0,1 at generation n, when started with a single ON cell, regarded as a binary number.

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A246028 a(n) = Product_{i in row n of A245562} Fibonacci(i+1).

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A246685 Run Length Transform of sequence 1, 3, 5, 17, 257, 65537, ... (1 followed by Fermat numbers).

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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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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.