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A270937
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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 221", based on the 5-celled von Neumann neighborhood.
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1
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7, -4, 40, -35, 107, -107, 211, -211, 347, -347, 515, -515, 715, -715, 947, -947, 1211, -1211, 1507, -1507, 1835, -1835, 2195, -2195, 2587, -2587, 3011, -3011, 3467, -3467, 3955, -3955, 4475, -4475, 5027, -5027, 5611, -5611, 6227, -6227, 6875, -6875, 7555
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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) = 4-9*(-1)^n+(4+8*(-1)^n)*n+4*(-1)^n*n^2 for n>2.
a(n) = 4*n^2+12*n-5 for n>2 and even.
a(n) = -4*n^2-4*n+13 for n>2 and odd.
a(n) = -a(n-1)+2*a(n-2)+2*a(n-3)-a(n-4)-a(n-5) for n>7.
G.f.: (7+3*x+22*x^2-x^3+7*x^4-7*x^5-4*x^6+5*x^7) / ((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=221; 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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