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A270334 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 158", based on the 5-celled von Neumann neighborhood. 0
1, 5, 20, 52, 236, 916, 4060, 16904, 66660, 265800, 1059724, 4219684, 16816424, 67131892, 268216416, 1072151796 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

REFERENCES

Stephen 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

Stephen 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

(* From Robert Price, Start *)

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

code = 158; 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 *)

(* From Robert Price, End *)

Total[#, 2] & /@ Array[CellularAutomaton[{158, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, {{{2^# - 1}}}] &, 6] (* JungHwan Min, Mar 16 2016 *)

CROSSREFS

Cf. A270333.

Sequence in context: A102227 A173034 A006010 * A008524 A055383 A090133

Adjacent sequences:  A270331 A270332 A270333 * A270335 A270336 A270337

KEYWORD

nonn,more

AUTHOR

Robert Price, Mar 15 2016

EXTENSIONS

a(8)-a(15) from Lars Blomberg, May 07 2016

STATUS

approved

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Last modified March 7 10:30 EST 2021. Contains 341869 sequences. (Running on oeis4.)