A253100 Number of odd terms in f^n, where f = 1/(x*y)+1/x+1/x*y+1/y+x+x*y.

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

Offset

0,2

Comments

This is the number of ON cells in a certain two-dimensional 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 odd-rule cellular automaton defined by OddRule 347 (see Ekhad-Sloane-Zeilberger "Odd-Rule 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 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

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 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*y)+1/x+1/x*y+1/y+x+x*y;
OddCA(f, 130);

Mathematica

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