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A270082
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First differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 62", based on the 5-celled von Neumann neighborhood.
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1
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4, 7, 8, 12, 12, 24, 4, 28, 12, 44, 4, 44, 12, 60, 4, 60, 12, 76, 4, 76, 12, 92, 4, 92, 12, 108, 4, 108, 12, 124, 4, 124, 12, 140, 4, 140, 12, 156, 4, 156, 12, 172, 4, 172, 12, 188, 4, 188, 12, 204, 4, 204, 12, 220, 4, 220, 12, 236, 4, 236, 12, 252, 4, 252
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OFFSET
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0,1
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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) = a(n-2)+a(n-4)-a(n-6) for n>9.
G.f.: (4+7*x+4*x^2+5*x^3+5*x^5-12*x^6-x^7+4*x^8+4*x^9-4*x^11) / ((1-x)^2*(1+x)^2*(1+x^2)).
(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=62; 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[on[[i+1]]-on[[i]], {i, 1, Length[on]-1}] (* Difference 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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