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A270106
Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 84", based on the 5-celled von Neumann neighborhood.
1
1, 5, 13, 29, 45, 77, 109, 173, 205, 269, 333, 461, 525, 653, 781, 1037, 1101, 1229, 1357, 1613, 1741, 1997, 2253, 2765, 2893, 3149, 3405, 3917, 4173, 4685, 5197, 6221, 6349, 6605, 6861, 7373, 7629, 8141, 8653, 9677, 9933, 10445, 10957, 11981, 12493, 13517
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=84; 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[Total[Part[on, Range[1, i]]], {i, 1, Length[on]}] (* Sum at each stage *)
CROSSREFS
Cf. A189007.
Sequence in context: A065374 A130066 A206258 * A304904 A095085 A309371
KEYWORD
nonn,easy
AUTHOR
Robert Price, Mar 11 2016
STATUS
approved