

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 2D 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 oddrule cellular automaton defined by OddRule 171 (see EkhadSloaneZeilberger "OddRule 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 MetaAlgorithm for Creating Fast Algorithms for Counting ON Cells in OddRule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, OddRule 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 n1 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[n5] + 6 f[n4] + 13 f[n3] + 5 f[n2] + f[n1]; Table[Times @@ (f[Length[#]]&) /@ Select[s = Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 67}] (* JeanFrançois 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



