A272991 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 569", based on the 5-celled von Neumann neighborhood.

1, 4, 17, 28, 61, 93, 133, 149, 213, 297, 353, 413, 497, 577, 685, 665, 781, 1029, 1093, 1245, 1381, 1473, 1701, 1757, 1929, 2189, 2281, 2417, 2633, 2741, 2941, 2965, 3153, 3649, 3757, 4177, 4377, 4553, 4977, 5113, 5377, 5693, 5901, 6145, 6653, 6773, 7081

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=569; 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: A273610 A272769 A273646 * A273430 A273073 A272330

Adjacent sequences: A272988 A272989 A272990 * A272992 A272993 A272994

Keyword

nonn,easy

Author

Robert Price, May 12 2016

Status

approved