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A269909
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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 1", based on the 5-celled von Neumann neighborhood.
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2
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3, -3, 43, -43, 115, -115, 219, -219, 355, -355, 523, -523, 723, -723, 955, -955, 1219, -1219, 1515, -1515, 1843, -1843, 2203, -2203, 2595, -2595, 3019, -3019, 3475, -3475, 3963, -3963, 4483, -4483, 5035, -5035, 5619, -5619, 6235, -6235, 6883, -6883, 7563
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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+(4+8*(-1)^n)*n+4*(-1)^n*n^2).
a(n) = 4*n^2+12*n+3 for n even.
a(n) = -4*n^2-4*n+5 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.: (3+34*x^2-5*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=1; stages=100;
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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