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A269906
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Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 1", based on the 5-celled von Neumann neighborhood.
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4
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1, 4, 1, 44, 1, 116, 1, 220, 1, 356, 1, 524, 1, 724, 1, 956, 1, 1220, 1, 1516, 1, 1844, 1, 2204, 1, 2596, 1, 3020, 1, 3476, 1, 3964, 1, 4484, 1, 5036, 1, 5620, 1, 6236, 1, 6884, 1, 7564, 1, 8276, 1, 9020, 1, 9796, 1, 10604, 1, 11444, 1, 12316, 1, 13220, 1
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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) = (-3+5*(-1)^n-4*(-1+(-1)^n)*n-4*(-1+(-1)^n)*n^2)/2.
a(n) = 1 for n even.
a(n) = 4*n^2+4*n-4 for n odd.
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>5.
G.f.: (1+4*x-2*x^2+32*x^3+x^4-4*x^5) / ((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=1; 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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