A255279 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y)-y/x-x*y.

1, 7, 7, 25, 7, 49, 25, 107, 7, 49, 49, 175, 25, 175, 107, 413, 7, 49, 49, 175, 49, 343, 175, 749, 25, 175, 175, 625, 107, 749, 413, 1615, 7, 49, 49, 175, 49, 343, 175, 749, 49, 343, 343, 1225, 175, 1225, 749, 2891, 25, 175, 175, 625, 175, 1225

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 277 (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 A255280.

Example

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

Mathematica

(* f = A255280 *) f[0]=1; f[1]=7; f[2]=25; f[3]=107; f[4]=413; f[5]=1615; f[n_] := f[n] = -16 f[n-7] - 16 f[n-6] + 16 f[n-4] - 9 f[n-3] - 3 f[n-2] + 5 f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[ Split[ IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 53}] (* Jean-François Alcover, Jul 12 2017 *)

Crossrefs

Cf. A255280.

Sequence in context: A290520 A363153 A247666 * A230496 A295733 A341201

Adjacent sequences: A255276 A255277 A255278 * A255280 A255281 A255282

Keyword

nonn

Author

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

Status

approved