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A273406
Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 673", based on the 5-celled von Neumann neighborhood.
1
1, 5, 26, 70, 147, 263, 428, 648, 933, 1289, 1726, 2250, 2871, 3595, 4432, 5388, 6473, 7693, 9058, 10574, 12251, 14095, 16116, 18320, 20717, 23313, 26118, 29138, 32383, 35859, 39576, 43540, 47761, 52245, 57002, 62038, 67363, 72983, 78908, 85144, 91701, 98585
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 22 2016: (Start)
a(n) = (9+3*(-1)^n-10*n+48*n^2+16*n^3)/12.
a(n) = (8*n^3+24*n^2-5*n+6)/6 for n even.
a(n) = (8*n^3+24*n^2-5*n+3)/6 for n odd.
a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5) for n>4.
G.f.: (1+2*x+13*x^2+4*x^3-4*x^4) / ((1-x)^4*(1+x)).
(End)
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=673; 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. A273405.
Sequence in context: A185939 A273419 A273447 * A273833 A273849 A273781
KEYWORD
nonn,easy
AUTHOR
Robert Price, May 21 2016
STATUS
approved