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A270081
Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 62", based on the 5-celled von Neumann neighborhood.
1
1, 6, 18, 38, 70, 114, 182, 254, 354, 466, 622, 782, 986, 1202, 1478, 1758, 2098, 2450, 2878, 3310, 3818, 4338, 4950, 5566, 6274, 6994, 7822, 8654, 9594, 10546, 11622, 12702, 13906, 15122, 16478, 17838, 19338, 20850, 22518, 24190, 26018, 27858, 29870, 31886
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.
FORMULA
Empirical g.f.: (1+4*x+6*x^2+4*x^3+4*x^4+6*x^6-12*x^7-x^8+4*x^9+4*x^10-4*x^12) / ((1-x)^4*(1+x)^2*(1+x^2)). - Colin Barker, Mar 10 2016
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=62; 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. A270079.
Sequence in context: A180118 A270335 A270940 * A261651 A270215 A129863
KEYWORD
nonn,easy
AUTHOR
Robert Price, Mar 10 2016
STATUS
approved