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A271061
Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 261", based on the 5-celled von Neumann neighborhood.
1
1, 8, 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.
Similar to A211012.
It is conjectured that Rules 269, 277, 285, 293, 301, 309, 317, 325, 333, 341, 349, 357, 365, 373, 381, 413, 445, 477, 509, 645, 653, 661, 669, 677, 685, 693, 701, 709, 717, 725, 733, 741, 749, 757 and 765 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
Conjecture: a(n) = 4*4^n - 4*2^n, n>0. - Lars Blomberg, Jun 09 2016
Conjectures from Colin Barker, Dec 01 2016: (Start)
a(n) = 6*a(n-1) - 8*a(n-2) for n>2.
G.f.: (1 + 2*x + 8*x^2) / ((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=261; 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: A263507 A261975 A087914 * A211012 A081084 A230931
KEYWORD
nonn,more
AUTHOR
Robert Price, Mar 29 2016
EXTENSIONS
a(8)-a(15) from Lars Blomberg, Jun 09 2016
STATUS
approved