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A271064
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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 261", based on the 5-celled von Neumann neighborhood.
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1
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7, -7, 47, -47, 119, -119, 223, -223, 359, -359, 527, -527, 727, -727, 959, -959, 1223, -1223, 1519, -1519, 1847, -1847, 2207, -2207, 2599, -2599, 3023, -3023, 3479, -3479, 3967, -3967, 4487, -4487, 5039, -5039, 5623, -5623, 6239, -6239, 6887, -6887, 7567
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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) = - a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4) - a(n-5) for n>4.
G.f.: (-x^4 + 26*x^2 + 7)/((x - 1)^2*(x + 1)^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=261; 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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