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A273335
Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 657", based on the 5-celled von Neumann neighborhood.
0
1, 4, 48, 224, 960, 3968, 16128, 65024, 261120, 1046528, 4190208, 16769024, 67092480, 268402688, 1073676288, 4294836224
OFFSET
0,2
COMMENTS
Initialized with a single black (ON) cell at stage zero.
Conjecture: Rule 665 also generates this sequence. - Lars Blomberg, Jul 18 2016
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
FORMULA
Conjectures from Colin Barker, May 20 2016: (Start)
a(n) = 2^(n+2)*(2^n-1) for n>1.
a(n) = 6*a(n-1)-8*a(n-2) for n>3.
G.f.: (1-2*x+32*x^2-32*x^3) / ((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=657; 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. A273334.
Sequence in context: A192418 A162673 A002287 * A275138 A273767 A292074
KEYWORD
nonn,more
AUTHOR
Robert Price, May 20 2016
EXTENSIONS
a(8)-a(15) from Lars Blomberg, Jul 18 2016
STATUS
approved