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A282609
Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 489", based on the 5-celled von Neumann neighborhood.
4
1, 0, 7, 2, 27, 30, 103, 90, 315, 94, 1255, 346, 4923, 1630, 19687, 6490, 78651, 26206, 314599, 104794, 1258299, 419422, 5033191, 1677658, 20132667, 6710878, 80530663, 26843482, 322122555, 107374174, 1288490215, 429496666, 5153960763, 1717986910, 20615843047
OFFSET
0,3
COMMENTS
Initialized with a single black (ON) cell at stage zero.
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
FORMULA
Conjectures from Chai Wah Wu, May 05 2024: (Start)
a(n) = 4*a(n-2) + a(n-4) - 4*a(n-6) for n > 13.
G.f.: (512*x^13 - 288*x^9 - 96*x^8 - 32*x^7 - 8*x^6 + 22*x^5 - 2*x^4 + 2*x^3 + 3*x^2 + 1)/(4*x^6 - x^4 - 4*x^2 + 1). (End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 489; 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}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 2], {i , 1, stages - 1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Feb 19 2017
STATUS
approved