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A272154 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 437", based on the 5-celled von Neumann neighborhood. 0
1, 8, 44, 220, 900, 3776, 15392, 62128, 250116, 1001704, 4007524, 16042116, 64173160, 256653648, 1026693944, 4106769456 (list; graph; refs; listen; history; text; internal format)



Initialized with a single black (ON) cell at stage zero.


S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.


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


CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];

code=437; 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=Table[ca=CAStep[rule, ca], {n, 1, stages+1}];

PrependTo[ca, a];

(* Trim full grid to reflect growth by one cell at each stage *)


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 *)


Cf. A272153.

Sequence in context: A125318 A000373 A176688 * A270935 A197213 A198768

Adjacent sequences:  A272151 A272152 A272153 * A272155 A272156 A272157




Robert Price, Apr 21 2016


a(8)-a(15) from Lars Blomberg, Jul 03 2016



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Last modified December 1 09:28 EST 2020. Contains 338833 sequences. (Running on oeis4.)