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A271004
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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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4
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1, 8, 4, 44, 13, 112, 25, 208, 37, 340, 49, 504, 61, 700, 73, 928, 85, 1188, 97, 1480, 109, 1804, 121, 2160, 133, 2548, 145, 2968, 157, 3420, 169, 3904, 181, 4420, 193, 4968, 205, 5548, 217, 6160, 229, 6804, 241, 7480, 253, 8188, 265, 8928, 277, 9700, 289
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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) = (-13-9*(-1)^n+4*(2+(-1)^n)*n-4*(-1+(-1)^n)*n^2)/2 for n>5.
a(n) = 6*n-11 for n>3 and even.
a(n) = 4*n^2+2*n-2 for n>5 and odd.
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) 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)^3*(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}];
Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells 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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