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A253066 Number of odd terms in f^n, where f = 1/x+1+x+1/y+y/x+x*y. 5
1, 6, 6, 28, 6, 36, 28, 112, 6, 36, 36, 168, 28, 168, 112, 456, 6, 36, 36, 168, 36, 216, 168, 672, 28, 168, 168, 784, 112, 672, 456, 1816, 6, 36, 36, 168, 36, 216, 168, 672, 36, 216, 216, 1008, 168, 1008, 672, 2736, 28, 168, 168, 784, 168, 1008, 784, 3136, 112, 672, 672, 3136, 456, 2736, 1816, 7288 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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 275 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).

LINKS

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 [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

FORMULA

This is the Run Length Transform of A253068.

EXAMPLE

Here is the neighborhood f:

[X, 0, X]

[X, X, X]

[0, X, 0]

which contains a(1) = 6 ON cells.

MAPLE

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/x+x*y;

OddCA(f, 130);

MATHEMATICA

(* f = A253068 *) f[0] = 1; f[n_] := ((-2)^n + 4^(n+2)-8)/9; Table[Times @@ (f[Length[#]]&) /@ Select[s = Split[IntegerDigits[n, 2]], #[[1]] == 1 &], {n, 0, 63}] (* Jean-Fran├žois Alcover, Jul 12 2017 *)

CROSSREFS

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

Cf. A253068.

Sequence in context: A241865 A243122 A274940 * A016725 A267651 A151779

Adjacent sequences:  A253063 A253064 A253065 * A253067 A253068 A253069

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 29 2015

STATUS

approved

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Last modified September 15 14:32 EDT 2019. Contains 327078 sequences. (Running on oeis4.)