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

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

Author

N. J. A. Sloane, Jan 29 2015

Keywords

• nonn

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

Examples

			Here is the neighborhood f:
[X, 0, X]
[X, X, X]
[0, 0, X]
which contains a(1) = 6 ON cells.
		

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

Crossrefs

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.

Programs

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+x/y+y/x+x*y;
OddCA(f, 130);

Mathematica

(* 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 *)

Formula

This is the Run Length Transform of A253070.