A253104 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y)-y.

1, 8, 8, 32, 8, 64, 32, 140, 8, 64, 64, 256, 32, 256, 140, 580, 8, 64, 64, 256, 64, 512, 256, 1120, 32, 256, 256, 1024, 140, 1120, 580, 2300, 8, 64, 64, 256, 64, 512, 256, 1120, 64, 512, 512, 2048, 256, 2048, 1120, 4640, 32, 256, 256, 1024, 256

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 577 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link).

Links

Table of n, a(n) for n=0..52.

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 A253105.

Example

Here is the neighborhood f:
[X, 0, X]
[X, X, X]
[X, X, X]
which contains a(1) = 8 ON cells.

Mathematica

(* f = A253105 *) f[0]=1; f[1]=8; f[2]=32; f[3]=140; f[4]=580; f[5]=2300; f[n_] := f[n] = -8f[n-7] + 8f[n-6] + 12f[n-5] - 29f[n-4] + 6f[n-3] + 4f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 52}] (* Jean-François Alcover, Jul 12 2017 *)

Crossrefs

Cf. A253105.

Sequence in context: A339734 A341834 A255275 * A122858 A143336 A328529

Adjacent sequences: A253101 A253102 A253103 * A253105 A253106 A253107

Keyword

nonn

Author

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

Status

approved