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A269913
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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 3", based on the 5-celled von Neumann neighborhood.
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1
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4, -4, 44, -44, 116, -116, 220, -220, 356, -356, 524, -524, 724, -724, 956, -956, 1220, -1220, 1516, -1516, 1844, -1844, 2204, -2204, 2596, -2596, 3020, -3020, 3476, -3476, 3964, -3964, 4484, -4484, 5036, -5036, 5620, -5620, 6236, -6236, 6884, -6884, 7564
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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) = 4*(1+n+2*(-1)^n*n+(-1)^n*n^2).
a(n) = 4*(1+n+2*1*n+1*n^2) for n even.
a(n) = 4*(1-n-n^2) for n odd.
a(n) = -a(n-1)+2*a(n-2)+2*a(n-3)-a(n-4)-a(n-5) for n>4.
G.f.: 4*(1+8*x^2-x^4) / ((1-x)^2*(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=3; 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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sign,easy
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AUTHOR
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STATUS
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approved
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