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A272051
Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 421", based on the 5-celled von Neumann neighborhood.
1
1, 9, 14, 54, 71, 179, 208, 388, 453, 777, 838, 1298, 1411, 2087, 2188, 2976, 3201, 4293, 4478, 5862, 6111, 7767, 8004, 9868, 10245, 12533, 12850, 15510, 15979, 19043, 19512, 22884, 23553, 27533, 28110, 32590, 33271, 38343, 39104, 44604, 45329, 51453, 52334
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=421; 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. A272049.
Sequence in context: A239038 A271596 A272275 * A272504 A346267 A197955
KEYWORD
nonn,easy
AUTHOR
Robert Price, Apr 18 2016
STATUS
approved