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A271813 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 405", based on the 5-celled von Neumann neighborhood. 0
1, 8, 44, 212, 916, 3736, 15372, 61728, 247980, 993524, 3978640, 15927748, 63725264, 254965564, 1019900556, 4079770028 (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

Table of n, a(n) for n=0..15.

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=405; 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}];

on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)

Part[on, 2^Range[0, Log[2, stages]]] (* Extract relevant terms *)

CROSSREFS

Cf. A271812.

Sequence in context: A270318 A270330 A265613 * A270678 A125318 A000373

Adjacent sequences:  A271810 A271811 A271812 * A271814 A271815 A271816

KEYWORD

nonn,more

AUTHOR

Robert Price, Apr 14 2016

EXTENSIONS

a(8)-a(15) from Lars Blomberg, Jun 29 2016

STATUS

approved

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Last modified June 25 06:13 EDT 2017. Contains 288709 sequences.