A271285 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 337", based on the 5-celled von Neumann neighborhood.

1, 4, 5, 40, 9, 100, 21, 200, 25, 324, 37, 480, 49, 644, 85, 872, 89, 1124, 101, 1408, 113, 1700, 149, 2048, 161, 2404, 197, 2792, 233, 3140, 341, 3624, 345, 4132, 357, 4672, 369, 5220, 405, 5824, 417, 6436, 453, 7080, 489, 7684, 597, 8416, 609, 9156, 645

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=337; 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: A270087 A265689 A270098 * A223528 A189744 A241279

Adjacent sequences: A271282 A271283 A271284 * A271286 A271287 A271288

Keyword

nonn,easy

Author

Robert Price, Apr 03 2016

Status

approved