A255277 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y)-y/x-y.

1, 7, 7, 27, 7, 49, 27, 113, 7, 49, 49, 189, 27, 189, 113, 447, 7, 49, 49, 189, 49, 343, 189, 791, 27, 189, 189, 729, 113, 791, 447, 1743, 7, 49, 49, 189, 49, 343, 189, 791, 49, 343, 343, 1323, 189, 1323, 791, 3129, 27, 189, 189, 729, 189, 1323, 729, 3051, 113, 791, 791, 3051, 447, 3129, 1743, 6789, 7

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 177 (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, 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 A255278.

Example

Here is the neighborhood f:
[0, 0, X]
[X, X, X]
[X, X, X]
which contains a(1) = 7 ON cells.

Mathematica

(* f = A255278 *) f[0]=1; f[1]=7; f[2]=27; f[3]=113; f[4]=447; f[5]=1743; f[6]=6789; f[n_] := f[n] = -12f[n-8] - 4f[n-7] + 6f[n-6] - 4f[n-5] - 11f[n-4] - 7f[n-3] + 6f[n-2] + 3f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 64}] (* Jean-François Alcover, Jul 12 2017 *)

Crossrefs

Cf. A255278.

Sequence in context: A230496 A295733 A341201 * A027844 A268867 A111217

Adjacent sequences: A255274 A255275 A255276 * A255278 A255279 A255280

Keyword

nonn

Author

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

Status

approved