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A253065 Number of odd terms in f^n, where f = 1+x+x^2+x^2*y+x^2/y. 5
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 (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 171 (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 A253067.

EXAMPLE

Here is the neighborhood f:

[0, 0, X]

[X, X, X]

[0, 0, X]

which contains a(1) = 5 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+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-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, A253066.

Cf. A253067.

Sequence in context: A273606 A273544 A273703 * A294612 A246332 A300784

Adjacent sequences:  A253062 A253063 A253064 * A253066 A253067 A253068

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jan 26 2015

STATUS

approved

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Last modified March 23 18:13 EDT 2019. Contains 321433 sequences. (Running on oeis4.)