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A285646
Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 65", based on the 5-celled von Neumann neighborhood.
4
1, 1, 4, 3, 16, 15, 96, 63, 384, 255, 1536, 1023, 6144, 4095, 24576, 16383, 98304, 65535, 393216, 262143, 1572864, 1048575, 6291456, 4194303, 25165824, 16777215, 100663296, 67108863, 402653184, 268435455, 1610612736, 1073741823, 6442450944, 4294967295
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 Colin Barker, Apr 24 2017: (Start)
G.f.: (1 + x - x^2 - 2*x^3 + 4*x^5 + 32*x^6 - 32*x^8) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
a(n) = 3*2^(n-1) for n>4 and even.
a(n) = 2^(n-1) - 1 for n>4 and odd.
a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 65; 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, Apr 23 2017
STATUS
approved