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A270084 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 65", based on the 5-celled von Neumann neighborhood. 1
1, 4, 36, 188, 876, 3788, 15756, 64268, 259596, 1043468, 4184076, 16756748, 67067916, 268353548, 1073577996, 4294639628 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

Similar to A271254.

Lars Blomberg conjectured that Rule 321 also produces this sequence.  It would be nice to have a proof.

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

FORMULA

Conjecture: a(n) = 4*4^n - 10*2^n + 12, n>1. - Lars Blomberg, Apr 21 2016

Conjectures from Colin Barker, Apr 21 2016: (Start)

a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>4.

G.f.: (1-3*x+22*x^2-16*x^3+32*x^4) / ((1-x)*(1-2*x)*(1-4*x)).

(End)

MATHEMATICA

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

code=65; 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. A269782, A271254.

Sequence in context: A276295 A272146 A034569 * A272222 A263420 A302946

Adjacent sequences:  A270081 A270082 A270083 * A270085 A270086 A270087

KEYWORD

nonn,more

AUTHOR

Robert Price, Mar 10 2016

EXTENSIONS

a(8)-a(15) from Lars Blomberg, Apr 21 2016

STATUS

approved

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Last modified April 9 14:02 EDT 2020. Contains 333353 sequences. (Running on oeis4.)