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A270098 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 81", based on the 5-celled von Neumann neighborhood. 4
1, 4, 5, 40, 9, 100, 21, 196, 29, 320, 41, 476, 53, 640, 89, 844, 117, 1096, 129, 1380, 141, 1672, 177, 2004, 197, 2368, 217, 2764, 245, 3128, 321, 3532, 405, 4056, 369, 4652, 341, 5216, 349, 5824, 361, 6412, 429, 7032, 477, 7596, 561, 8324, 605, 9064, 593 (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

Robert Price, Diagrams for 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=81; 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: A041557 A270087 A265689 * A271285 A223528 A189744

Adjacent sequences:  A270095 A270096 A270097 * A270099 A270100 A270101

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 11 2016

STATUS

approved

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Last modified May 9 07:56 EDT 2021. Contains 343692 sequences. (Running on oeis4.)