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A271006
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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 245", based on the 5-celled von Neumann neighborhood.
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1
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1, 9, 13, 57, 70, 182, 207, 415, 452, 792, 841, 1345, 1406, 2106, 2179, 3107, 3192, 4380, 4477, 5957, 6066, 7870, 7991, 10151, 10284, 12832, 12977, 15945, 16102, 19522, 19691, 23595, 23776, 28196, 28389, 33357, 33562, 39110, 39327, 45487, 45716, 52520, 52761
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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*(71-7*(-1)^n)-(25*n)/6-(-3+(-1)^n)*n^2+(2*n^3)/3 for n>4.
a(n) = (4*n^3+12*n^2-25*n+96)/6 for n>4 and even.
a(n) = (4*n^3+24*n^2-25*n+117)/6 for n>4 and 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>9.
G.f.: (1+8*x+x^2+20*x^3+4*x^4+4*x^5-3*x^6-4*x^7-3*x^8+8*x^9-4*x^11) / ((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=245; 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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