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A273447
Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 721", based on the 5-celled von Neumann neighborhood.
1
1, 5, 26, 67, 147, 267, 435, 659, 947, 1307, 1747, 2275, 2899, 3627, 4467, 5427, 6515, 7739, 9107, 10627, 12307, 14155, 16179, 18387, 20787, 23387, 26195, 29219, 32467, 35947, 39667, 43635, 47859, 52347, 57107, 62147, 67475, 73099, 79027, 85267, 91827, 98715
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
Conjectures from Colin Barker, May 23 2016: (Start)
a(n) = (4*n^3+12*n^2+8*n-39)/3 for n>2.
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>6.
G.f.: (1+x+12*x^2-11*x^3+16*x^4-18*x^5+7*x^6) / (1-x)^4.
(End)
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=721; 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. A273443.
Sequence in context: A139273 A185939 A273419 * A273406 A273833 A273849
KEYWORD
nonn,easy
AUTHOR
Robert Price, May 22 2016
STATUS
approved