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A283350
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 617", based on the 5-celled von Neumann neighborhood.
4
1, 0, 6, 8, 14, 24, 62, 120, 254, 504, 1022, 2040, 4094, 8184, 16382, 32760, 65534, 131064, 262142, 524280, 1048574, 2097144, 4194302, 8388600, 16777214, 33554424, 67108862, 134217720, 268435454, 536870904, 1073741822, 2147483640, 4294967294, 8589934584
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, Mar 06 2017: (Start)
G.f.: (1 - 2*x + 5*x^2 - 2*x^3 - 8*x^4 + 16*x^6) / ((1 - x)*(1 + x)*(1 - 2*x)).
a(n) = 2^n - 2 for n>3 and even.
a(n) = 2^n - 8 for n>3 and odd.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>4.
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 617; 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, Mar 05 2017
STATUS
approved