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A270012
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Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 7", based on the 5-celled von Neumann neighborhood.
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1
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1, 10, 10, 59, 59, 180, 180, 405, 405, 766, 766, 1295, 1295, 2024, 2024, 2985, 2985, 4210, 4210, 5731, 5731, 7580, 7580, 9789, 9789, 12390, 12390, 15415, 15415, 18896, 18896, 22865, 22865, 27354, 27354, 32395, 32395, 38020, 38020, 44261, 44261, 51150, 51150
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OFFSET
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0,2
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COMMENTS
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Initialized with a single black (ON) cell at stage zero.
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
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LINKS
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FORMULA
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a(n) = 1/4*(7-3*(-1)^n)+(11/6-2*(-1)^n)*n-(-2+(-1)^n)*n^2+(2*n^3)/3.
a(n) = (4*n^3+6*n^2-n+6)/6 for n even.
a(n) = (4*n^3+18*n^2+23*n+15)/6 for n odd.
a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7) for n>6.
G.f.: (1+9*x-3*x^2+22*x^3+3*x^4+x^5-x^6) / ((1-x)^4*(1+x)^3).
(End)
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MATHEMATICA
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CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=7; 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 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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