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A270167
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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 107", based on the 5-celled von Neumann neighborhood.
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1
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4, 0, 36, -29, 97, -93, 181, -157, 281, -261, 409, -385, 561, -533, 737, -705, 937, -901, 1161, -1121, 1409, -1365, 1681, -1633, 1977, -1925, 2297, -2241, 2641, -2581, 3009, -2945, 3401, -3333, 3817, -3745, 4257, -4181, 4721, -4641, 5209, -5125, 5721, -5633
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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) = 9/2*(1+(-1)^n)+(4+6*(-1)^n)*n+3*(-1)^n*n^2 for n>7.
a(n) = 3*n^2+10*n+9 for n>7 and even.
a(n) = -3*n^2-2*n for n>7 and odd.
a(n) = -a(n-1)+2*a(n-2)+2*a(n-3)-a(n-4)-a(n-5) for n>12.
G.f.: (4+4*x+28*x^2-x^3-6*x^5-12*x^6+23*x^7+16*x^8-24*x^9-12*x^10+8*x^11+4*x^12) / ((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=107; 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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