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 377 (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 A255276.
EXAMPLE
Here is the neighborhood f:
[0, X, X]
[X, X, X]
[X, X, X]
which contains a(1) = 8 ON cells.
MATHEMATICA
(* f = A255276 *) f[0]=1; f[1]=8; f[2]=28; f[3]=128; f[4]=480; f[5]=2008; f[6]=7776; f[n_] := f[n] = -32f[n-8] - 24f[n-7] + 164f[n-6] - 236f[n-5] + 145f[n-4] - 24f[n-3] - 16f[n-2] + 8f[n-1]; Table[Times @@ (f[Length[#]]&) /@ Select[ Split[ IntegerDigits[n, 2]], #[[1]] == 1&], {n, 0, 52}] (* Jean-François Alcover, Jul 12 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane and Doron Zeilberger, Feb 19 2015
STATUS
approved