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A282417
Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 469", based on the 5-celled von Neumann neighborhood.
4
1, 2, 3, 12, 7, 56, 15, 240, 31, 992, 63, 4032, 127, 16256, 255, 65280, 511, 261632, 1023, 1047552, 2047, 4192256, 4095, 16773120, 8191, 67100672, 16383, 268419072, 32767, 1073709056, 65535, 4294901760, 131071, 17179738112, 262143, 68719214592, 524287
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
Conjectures from Colin Barker, Feb 15 2017: (Start)
a(n) = 7*a(n-2) - 14*a(n-4) + 8*a(n-6) for n>5.
G.f.: (1 + 2*x - 4*x^2 - 2*x^3) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 - 2*x^2)).
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 469; 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[1, i]], 2], {i, 1, stages - 1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Feb 14 2017
STATUS
approved