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A270021
First differences of 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.
1
4, 0, 32, -28, 100, -96, 200, -200, 320, -304, 464, -460, 652, -616, 808, -800, 1056, -1040, 1328, -1336, 1640, -1600, 1944, -1932, 2320, -2316, 2696, -2672, 3128, -3112, 3576, -3548, 4000, -3976, 4500, -4444, 4996, -4976, 5528, -5528, 6172, -6076, 6768
OFFSET
0,1
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}];
on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
Table[on[[i+1]]-on[[i]], {i, 1, Length[on]-1}] (* Difference at each stage *)
CROSSREFS
Cf. A270018.
Sequence in context: A271057 A269879 A269817 * A270328 A270933 A270293
KEYWORD
sign,easy
AUTHOR
Robert Price, Mar 08 2016
STATUS
approved