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A273420
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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 705", based on the 5-celled von Neumann neighborhood.
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1
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3, 17, 20, 31, 41, 47, 57, 63, 73, 79, 89, 95, 105, 111, 121, 127, 137, 143, 153, 159, 169, 175, 185, 191, 201, 207, 217, 223, 233, 239, 249, 255, 265, 271, 281, 287, 297, 303, 313, 319, 329, 335, 345, 351, 361, 367, 377, 383, 393, 399, 409, 415, 425, 431
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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) = 8+(-1)^n+8*n for n>2.
a(n) = 8*n+9 for n>2 and even.
a(n) = 8*n+7 for n>2 and odd.
a(n) = a(n-1)+a(n-2)-a(n-3) for n>5.
G.f.: (3+14*x-3*x^3+7*x^4-5*x^5) / ((1-x)^2*(1+x)).
(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=705; 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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nonn,easy
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AUTHOR
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STATUS
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approved
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