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A270126
Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 86", based on the 5-celled von Neumann neighborhood.
0
1, 5, 24, 96, 384, 1536, 6144, 24576, 98304, 393216, 1572864, 6291456, 25165824, 100663296, 402653184, 1610612736
OFFSET
0,2
COMMENTS
Initialized with a single black (ON) cell at stage zero.
Lars Blomberg conjectured that Rule 342 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.
FORMULA
Conjectures from Colin Barker, Mar 21 2016: (Start)
a(n) = 4*a(n-1) for n>2.
a(n) = 3*2^(2*n-1) for n>1.
G.f.: (1+x+4*x^2) / (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=86; 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
Sequence in context: A268370 A087095 A099653 * A276139 A078820 A291395
KEYWORD
nonn,more
AUTHOR
Robert Price, Mar 11 2016
EXTENSIONS
a(8)-a(15) from Lars Blomberg, Apr 23 2016
STATUS
approved