A246035 -id:A246035 - cp's OEIS frontend

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

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.

A253102 a(n) = A071053(n)^3.

Original entry on oeis.org

1, 27, 27, 125, 27, 729, 125, 1331, 27, 729, 729, 3375, 125, 3375, 1331, 9261, 27, 729, 729, 3375, 729, 19683, 3375, 35937, 125, 3375, 3375, 15625, 1331, 35937, 9261, 79507, 27, 729, 729, 3375, 729, 19683, 3375, 35937, 729, 19683, 19683, 91125, 3375, 91125, 35937, 250047, 125, 3375, 3375, 15625

Offset: 0

N. J. A. Sloane, Feb 20 2015

nonn

Number of ON cells at n-th generation of 3-D CA defined by generalization of Rule 150, starting with a single ON cell at generation 0.
Number of odd coefficients in ((1/x+1+x)*(1/y+1+y)*(1/z+1+z))^n.
Run length transform of A253103.

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

Cf. A071053, A246035, A253103.

a71053[n_] := Total[CoefficientList[(x^2 + x + 1)^n, x, Modulus -> 2]];
Table[a71053[n]^3, {n, 0, 51}] (* Jean-François Alcover, Sep 15 2018 *)

A383369 Population of elementary triangular automaton rule 90 at generation n, starting from a lone 1 cell at generation 0.

Original entry on oeis.org

1, 4, 6, 12, 6, 24, 24, 48, 6, 24, 36, 72, 24, 96, 96, 192, 6, 24, 36, 72, 36, 144, 144, 288, 24, 96, 144, 288, 96, 384, 384, 768, 6, 24, 36, 72, 36, 144, 144, 288, 36, 144, 216, 432, 144, 576, 576, 1152, 24, 96, 144, 288, 144, 576, 576, 1152, 96, 384, 576, 1152, 384, 1536, 1536, 3072, 6

Offset: 0

Paul Cousin, Apr 24 2025

nonn

An Elementary Triangular Automaton (ETA) is a cellular automaton in the triangular grid where cells hold binary states and rules are local to the first neighborhood. There are 256 possible ETA rules.
Rule 90 (1011010 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            |0|1|0|1|1|0|1|0|
  -----------------------------------------------
This is one of the 4 ETA rules (85, 90, 165 and 170) that replicates the pattern given as initial condition.

			Written as an irregular triangle with row lengths A000079, starting from n=1, the sequence begins:
  4;
  6, 12;
  6, 24, 24, 48;
  6, 24, 36, 72, 24, 96, 96, 192;
  6, 24, 36, 72, 36, 144, 144, 288, 24, 96, 144, 288, 96, 384, 384, 768;
...
It appears that the right border gives A110594.

Paul Cousin, Table of n, a(n) for n = 0..16384
Paul Cousin, Illustration for n = 0..128
Paul Cousin, Triangular Automata
Paul Cousin, Rule 90
Paul Cousin, Triangular Automata Integer Sequences
Paul Cousin, Triangular Automata: The 256 Elementary Cellular Automata of the Two-Dimensional Plane, Complex Systems, 33(3), 2024, pp. 253-276.

Pattern replicating ETA rules: A275667 (rule 170).
A247640 is a bisection.
A246035 is the analog on the square cells.

cp's OEIS Frontend

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

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A253102 a(n) = A071053(n)^3.

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A383369 Population of elementary triangular automaton rule 90 at generation n, starting from a lone 1 cell at generation 0.

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cp's OEIS Frontend

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

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Maple

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Formula

A253102 a(n) = A071053(n)^3.

Views

Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

A383369 Population of elementary triangular automaton rule 90 at generation n, starting from a lone 1 cell at generation 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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