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A270731
Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 205", based on the 5-celled von Neumann neighborhood.
1
1, 9, 13, 53, 70, 178, 195, 395, 428, 740, 785, 1273, 1362, 1982, 2087, 2979, 3084, 4128, 4281, 5617, 5798, 7438, 7666, 9627, 9895, 12160, 12444, 15181, 15605, 18646, 19022, 22618, 23086, 27007, 27615, 31988, 32708, 37644, 38465, 44001, 44766, 50702, 51647
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=205; 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. A270729.
Sequence in context: A270459 A270948 A270284 * A270467 A270982 A323797
KEYWORD
nonn,easy
AUTHOR
Robert Price, Mar 22 2016
STATUS
approved