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A283357
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 621", based on the 5-celled von Neumann neighborhood.
4
1, 2, 4, 11, 17, 46, 70, 190, 286, 766, 1150, 3070, 4606, 12286, 18430, 49150, 73726, 196606, 294910, 786430, 1179646, 3145726, 4718590, 12582910, 18874366, 50331646, 75497470, 201326590, 301989886, 805306366, 1207959550, 3221225470, 4831838206, 12884901886
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, Mar 07 2017: (Start)
G.f.: (1 + x - 2*x^2 + 3*x^3 - 2*x^4 + x^5 + 4*x^7) / ((1 - x)*(1 - 2*x)*(1 + 2*x)).
a(n) = (-32 - 3*(-2)^n + 21*2^n) / 16 for n>4.
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n>5.
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 621; 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, Mar 05 2017
STATUS
approved