A272012 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 414", based on the 5-celled von Neumann neighborhood.

1, 5, 12, 21, 32, 40, 81, 52, 96, 108, 193, 132, 217, 196, 329, 220, 340, 328, 512, 369, 505, 524, 736, 488, 852, 641, 984, 828, 1029, 908, 1405, 976, 1289, 1196, 1713, 1260, 1673, 1373, 1960, 1508, 2124, 1732, 2440, 1889, 2608, 2116, 2896, 2364, 3056, 2568

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

Robert Price, Diagrams of the first 20 stages

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

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=414; 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: A259913 A224824 A121291 * A272451 A097984 A028347

Adjacent sequences: A272009 A272010 A272011 * A272013 A272014 A272015

Keyword

nonn,easy

Author

Robert Price, Apr 17 2016

Status

approved