A253065 Number of odd terms in f^n, where f = 1+x+x^2+x^2*y+x^2/y.

1, 5, 5, 17, 5, 25, 17, 65, 5, 25, 25, 85, 17, 85, 65, 229, 5, 25, 25, 85, 25, 125, 85, 325, 17, 85, 85, 289, 65, 325, 229, 813, 5, 25, 25, 85, 25, 125, 85, 325, 25, 125, 125, 425, 85, 425, 325, 1145, 17, 85, 85, 289, 85, 425, 289, 1105, 65, 325, 325, 1105, 229, 1145, 813, 2945, 5, 25, 25, 85

Offset: 0

Author

N. J. A. Sloane, Jan 26 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 171 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).

Examples

			Here is the neighborhood f:
[0, 0, X]
[X, X, X]
[0, 0, X]
which contains a(1) = 5 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 [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

Crossrefs

Other CA's that use the same rule but with different cell neighborhoods: A160239, A102376, A071053, A072272, A001316, A246034, A246035, A253064, A253066.

Cf. A253067.

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+x^2+x^2*y+x^2/y;
OddCA(f, 130);

Mathematica

(* f = A253067 *) f[0]=1; f[1]=5; f[2]=17; f[3]=65; f[4]=229; f[5]=813; f[n_] := f[n] = 8 f[n-5] + 6 f[n-4] + 13 f[n-3] + 5 f[n-2] + f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[s = Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 67}] (* Jean-François Alcover, Jul 12 2017 *)

Formula

This is the Run Length Transform of A253067.