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A270091 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 78", based on the 5-celled von Neumann neighborhood. 4
1, 5, 8, 20, 20, 32, 48, 56, 88, 56, 96, 84, 140, 128, 180, 272, 260, 176, 312, 240, 328, 280, 408, 400, 552, 308, 536, 476, 748, 560, 848, 784, 1064, 768, 1092, 828, 1044, 928, 1068, 1196, 1100, 1124, 1332, 1408, 1660, 1472, 1560, 1716, 2180, 1684, 2056 (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 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=78; 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: A271095 A194266 A169703 * A270733 A270168 A270984

Adjacent sequences:  A270088 A270089 A270090 * A270092 A270093 A270094

KEYWORD

nonn,easy

AUTHOR

Robert Price, Mar 10 2016

STATUS

approved

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Last modified April 21 19:16 EDT 2021. Contains 343156 sequences. (Running on oeis4.)