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 367 (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
FORMULA
This is the Run Length Transform of A255282.
EXAMPLE
Here is the neighborhood f:
[0, X, X]
[X, X, 0]
[X, X, X]
which contains a(1) = 7 ON cells.
MATHEMATICA
(* f = A255282 *) f[0]=1; f[1]=7; f[2]=31; f[3]=127; f[4]=511; f[5]=2031; f[6]=8043; f[7]=31735; f[8]=125063; f[n_] := f[n] = 10 f[n-10] + 22 f[n-9] - 11 f[n-8] + 31 f[n-7] - 24 f[n-6] + 3 f[n-5] + 18 f[n-4] - 21 f[n-3] + 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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane and Doron Zeilberger, Feb 19 2015
STATUS
approved