A255283 Number of odd terms in f^n, where f = (1/x+1+x)*(1/y+1+y)-x-y.

1, 7, 7, 31, 7, 49, 31, 145, 7, 49, 49, 217, 31, 217, 145, 601, 7, 49, 49, 217, 49, 343, 217, 1015, 31, 217, 217, 961, 145, 1015, 601, 2551, 7, 49, 49, 217, 49, 343, 217, 1015, 49, 343, 343, 1519, 217, 1519, 1015, 4207, 31, 217, 217, 961, 217, 1519, 961, 4495, 145, 1015, 1015, 4495, 601, 4207, 2551, 10351

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 537 (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 A255284.

Example

Here is the neighborhood f:
[X, 0, X]
[X, X, 0]
[X, X, X]
which contains a(1) = 7 ON cells.
From Omar E. Pol, Feb 22 2015: (Start)
Written as an irregular triangle in which row lengths are the terms of A011782:
1;
7;
7, 31;
7, 49, 31, 145;
7, 49, 49, 217, 31, 217, 145, 601;
7, 49, 49, 217, 49, 343, 217, 1015, 31, 217, 217, 961, 145, 1015, 601, 2551;
...
Right border gives: 1, 7, 31, 145, 601, 2551, ... This is simply a restatement of the theorem that this sequence is the Run Length Transform of A255284.
(End)
From Omar E. Pol, Mar 19 2015: (Start)
Also, the sequence can be written as an irregular tetrahedron T(s,r,k) as shown below:
1;
..
7;
..
7;
31;
..........
7,     49;
31;
145;
......................
7,     49,   49,  217;
31,   217;
145;
601;
............................................
7,     49,   49,  217,  49,  343, 217, 1015;
31,   217,  217,  961;
145, 1015;
601;
2551;
.......................................................................................
7,     49,   49,  217,  49,  343, 217, 1015, 49, 343, 343, 1519, 217, 1519, 1015, 4207;
31,   217,  217,  961, 217, 1519, 961, 4495;
145, 1015, 1015, 4495;
601, 4207;
2551;
10351;
...
Apart from the initial 1, we have that T(s,r,k) = T(s+1,r,k).
(End)

Mathematica

(* f = A255284 *) f[n_] := If[EvenQ[n], 2^(2n+3)-5*7^(n/2), 2^(2n+3)-11*7^((n-1)/2)]/3; Table[Times @@ (f[Length[#]]&) /@ Select[Split[IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 63}] (* Jean-François Alcover, Jul 12 2017 *)

Crossrefs

Cf. A255284.

Sequence in context: A186142 A188274 A255281 * A140252 A095343 A286830

Adjacent sequences: A255280 A255281 A255282 * A255284 A255285 A255286

Keyword

nonn

Author

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

Status

approved