A270214 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 126", based on the 5-celled von Neumann neighborhood.

1, 5, 12, 20, 36, 40, 80, 52, 100, 104, 200, 104, 200, 208, 408, 116, 228, 232, 456, 232, 456, 464, 912, 232, 456, 464, 912, 464, 912, 928, 1832, 244, 484, 488, 968, 488, 968, 976, 1936, 488, 968, 976, 1936, 976, 1936, 1952, 3872, 488, 968, 976, 1936, 976

Offset

0,2

Comments

Initialized with a single black (ON) cell at stage zero.

References

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Robert Price, Table of n, a(n) for n = 0..128

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

S. Wolfram, A New Kind of Science

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

Robert Price, Diagrams of the first 20 stages.

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=126; stages=128;
rule=IntegerDigits[code, 2, 10];
g=2*stages+1; (* Maximum size of grid *)
a=PadLeft[{{1}}, {g, g}, 0, Floor[{g, g}/2]]; (* Initial ON cell on grid *)
ca=a;
ca=Table[ca=CAStep[rule, ca], {n, 1, stages+1}];
PrependTo[ca, a];
(* Trim full grid to reflect growth by one cell at each stage *)
k=(Length[ca[[1]]]+1)/2;
ca=Table[Table[Part[ca[[n]][[j]], Range[k+1-n, k-1+n]], {j, k+1-n, k-1+n}], {n, 1, k}];
Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)

Crossrefs

Sequence in context: A270938 A270079 A266937 * A063559 A259913 A224824

Adjacent sequences: A270211 A270212 A270213 * A270215 A270216 A270217

Keyword

nonn,easy

Author

Robert Price, Mar 13 2016

Status

approved