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A288137
Decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 427", based on the 5-celled von Neumann neighborhood.
4
1, 3, 3, 14, 7, 63, 14, 251, 31, 1022, 59, 4079, 126, 16379, 239, 65470, 507, 262127, 958, 1048315, 2031, 4194238, 3835, 16776175, 8126, 67108603, 15343, 268431294, 32507, 1073740783, 61374, 4294950651, 130031, 17179865022, 245499, 68719410159, 520126
OFFSET
0,2
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
G.f.: (1 + 3*x - x^2 + x^3 - 12*x^4 - 4*x^5 - 12*x^6) / ((1 - x)*(1 - 2*x)*(1 + 2*x)*(1 + x + x^2)*(1 - 2*x^2)*(1 + 2*x^2)) (conjectured). - Colin Barker, Jun 06 2017
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 427; 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]], 10], {i, 1, stages - 1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Jun 05 2017
STATUS
approved