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A270206 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 121", based on the 5-celled von Neumann neighborhood. 4
1, 4, 9, 32, 17, 100, 40, 164, 41, 300, 89, 440, 76, 636, 133, 800, 173, 1088, 193, 1288, 265, 1600, 348, 1880, 373, 2324, 344, 2649, 564, 2985, 564, 3505, 640, 3953, 756, 4349, 716, 5053, 716, 5497, 808, 6149, 980, 6645, 1112, 7305, 1300, 7885, 1416, 8489 (list; graph; refs; listen; history; text; internal format)
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=121; 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: A271414 A270177 A268277 * A271461 A272423 A280163

Adjacent sequences:  A270203 A270204 A270205 * A270207 A270208 A270209

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 13 2016

STATUS

approved

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Last modified June 19 15:03 EDT 2021. Contains 345141 sequences. (Running on oeis4.)