A272560 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 499", based on the 5-celled von Neumann neighborhood.

1, 5, 13, 36, 37, 88, 85, 156, 109, 268, 181, 380, 237, 564, 365, 740, 317, 1028, 393, 1360, 393, 1500, 781, 1600, 861, 2060, 993, 2420, 1081, 2824, 1481, 3024, 1449, 3816, 1325, 4440, 1513, 4692, 1881, 5052, 2117, 5648, 2273, 6284, 2621, 6620, 3041, 7484

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=499; 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: A264080 A290588 A272149 * A266102 A332599 A331453

Adjacent sequences: A272557 A272558 A272559 * A272561 A272562 A272563

Keyword

nonn,easy

Author

Robert Price, May 02 2016

Status

approved