A270988 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 241", based on the 5-celled von Neumann neighborhood.

1, 4, 13, 36, 37, 88, 77, 156, 113, 284, 121, 432, 229, 560, 337, 736, 305, 1072, 361, 1188, 601, 1452, 677, 1768, 853, 2144, 765, 2500, 1121, 2828, 1237, 3132, 1269, 3836, 1221, 4080, 1785, 4552, 1644, 5164, 2045, 5592, 2228, 6304, 2477, 6684, 2913, 7224

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=241; 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: A271358 A306349 A268996 * A272556 A173723 A002727

Adjacent sequences: A270985 A270986 A270987 * A270989 A270990 A270991

Keyword

nonn,easy

Author

Robert Price, Mar 27 2016

Status

approved