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

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

0,2

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 357 (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, 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

Formula

This is the Run Length Transform of A253072.

Example

Here is the neighborhood f:
[0, X, X]
[X, 0, X]
[X, X, X]
which contains a(1) = 7 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*y)+1/x+1/y+y+x/y+x+x*y;
OddCA(f, 130);

Mathematica

(* 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 *)

Crossrefs

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.

Sequence in context: A282828 A339722 A341833 * A283916 A146804 A147056

Adjacent sequences: A253068 A253069 A253070 * A253072 A253073 A253074

Keyword

nonn

Author

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

Status

approved