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A272703
Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 513", based on the 5-celled von Neumann neighborhood.
1
1, 5, 18, 42, 95, 160, 273, 385, 614, 855, 1176, 1465, 1970, 2459, 3060, 3540, 4505, 5482, 6603, 7628, 9061, 10414, 11943, 13160, 15313, 17418, 19731, 21764, 24517, 27006, 29735, 31719, 35692, 39677, 43934, 47967, 52792, 57409, 62330, 66555, 72612, 78493
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=513; 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. A272702.
Sequence in context: A000338 A212343 A056640 * A272736 A273532 A160969
KEYWORD
nonn,easy
AUTHOR
Robert Price, May 04 2016
STATUS
approved