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A282123
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 430", based on the 5-celled von Neumann neighborhood.
4
1, 3, 6, 15, 26, 63, 106, 255, 426, 1023, 1706, 4095, 6826, 16383, 27306, 65535, 109226, 262143, 436906, 1048575, 1747626, 4194303, 6990506, 16777215, 27962026, 67108863, 111848106, 268435455, 447392426, 1073741823, 1789569706, 4294967295, 7158278826
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 07 2017: (Start)
a(n) = (-5 - (-2)^n + (-1)^n + 11*2^n) / 6.
a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
G.f.: (1 + 3*x + x^2) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 430; 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 06 2017
STATUS
approved