A269872 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 41", based on the 5-celled von Neumann neighborhood.

1, 4, 5, 32, 13, 84, 29, 168, 33, 268, 61, 416, 53, 580, 121, 748, 105, 964, 185, 1188, 181, 1500, 221, 1784, 221, 2108, 305, 2480, 317, 2812, 437, 3188, 421, 3616, 545, 4152, 469, 4576, 645, 5124, 521, 5712, 813, 6100, 801, 6784, 885, 7356, 917, 8076, 1005

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..100

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

code=41; stages=100;
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: A271125 A270014 A271160 * A270075 A271148 A271200

Adjacent sequences: A269869 A269870 A269871 * A269873 A269874 A269875

Keyword

nonn,easy

Author

Robert Price, Mar 06 2016

Status

approved