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A270018
Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 51", based on the 5-celled von Neumann neighborhood.
4
1, 5, 5, 37, 9, 109, 13, 213, 13, 333, 29, 493, 33, 685, 69, 877, 77, 1133, 93, 1421, 85, 1725, 125, 2069, 137, 2457, 141, 2837, 165, 3293, 181, 3757, 209, 4209, 233, 4733, 289, 5285, 309, 5837, 309, 6481, 405, 7173, 325, 7789, 413, 8525, 457, 9237, 421
OFFSET
0,2
COMMENTS
Initialized with a single black (ON) cell at stage zero.
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=51; 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}];
Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
CROSSREFS
Sequence in context: A270325 A270930 A269814 * A269876 A139386 A074947
KEYWORD
nonn,easy
AUTHOR
Robert Price, Mar 08 2016
STATUS
approved