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
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
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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane and Doron Zeilberger, Feb 19 2015
STATUS
approved