

A253100


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


1



1, 6, 6, 24, 6, 36, 24, 96, 6, 36, 36, 144, 24, 144, 96, 372, 6, 36, 36, 144, 36, 216, 144, 576, 24, 144, 144, 576, 96, 576, 372, 1416, 6, 36, 36, 144, 36, 216, 144, 576, 36, 216, 216, 864, 144, 864, 576, 2232, 24, 144, 144, 576, 144, 864, 576, 2304, 96, 576, 576, 2304, 372, 2232, 1416, 5340
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OFFSET

0,2


COMMENTS

This is the number of ON cells in a certain twodimensional cellular automaton in which the neighborhood of a cell is defined by f, and in which a cell is ON iff there were an odd number of ON cells in the neighborhood at the previous generation.
This is the oddrule cellular automaton defined by OddRule 347 (see EkhadSloaneZeilberger "OddRule Cellular Automata on the Square Grid" link).


LINKS

Table of n, a(n) for n=0..63.
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, 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, 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 A253101.


EXAMPLE

Here is the neighborhood f:
[X, 0, X]
[X, 0, X]
[X, X, 0]
which contains a(1) = 6 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*y)+1/x+1/x*y+1/y+x+x*y;
OddCA(f, 130);


MATHEMATICA

(* f = A253101 *) f[n_] := 2*(2Sqrt[3])^n + 2*(2+Sqrt[3])^n  2^n // Round; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* JeanFrançois Alcover, Jul 12 2017 *)


CROSSREFS

Cf. A253101. Similar to but different from A247640.
Sequence in context: A255473 A255295 A255475 * A247640 A255470 A267710
Adjacent sequences: A253097 A253098 A253099 * A253101 A253102 A253103


KEYWORD

nonn


AUTHOR

N. J. A. Sloane and Doron Zeilberger, Feb 19 2015


STATUS

approved



