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A270132
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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 89", based on the 5-celled von Neumann neighborhood.
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1
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3, 1, 39, -44, 121, -121, 225, -225, 361, -361, 529, -529, 729, -729, 961, -961, 1225, -1225, 1521, -1521, 1849, -1849, 2209, -2209, 2601, -2601, 3025, -3025, 3481, -3481, 3969, -3969, 4489, -4489, 5041, -5041, 5625, -5625, 6241, -6241, 6889, -6889, 7569
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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+5*(-1)^n+(4+8*(-1)^n)*n+4*(-1)^n*n^2 for n>3.
a(n) = 4*n^2+12*n+9 for n>3 and even.
a(n) = -4*n^2-4*n-1 for n>3 and n odd.
a(n) = -a(n-1)+2*a(n-2)+2*a(n-3)-a(n-4)-a(n-5) for n>8.
G.f.: (3+4*x+34*x^2-13*x^3+14*x^5-10*x^6-5*x^7+5*x^8) / ((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=89; 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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