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A279876
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 211", based on the 5-celled von Neumann neighborhood.
4
1, 2, 5, 14, 1, 58, 21, 190, 257, 1018, 21, 4030, 1, 16378, 21, 65470, 1, 262138, 21, 1048510, 1, 4194298, 21, 16777150, 1, 67108858, 21, 268435390, 1, 1073741818, 21, 4294967230, 1, 17179869178, 21, 68719476670, 1, 274877906938, 21, 1099511627710, 1
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, Dec 21 2016: (Start)
a(n) = 4*a(n-2) + a(n-4) - 4*a(n-6) for n>10.
G.f.: (1 +2*x +x^2 +6*x^3 -20*x^4 +16*x^6 -48*x^7 +192*x^8 +256*x^9 -1024*x^10 -256*x^12 +1024*x^14) / ((1 -x)*(1 +x)*(1 -2*x)*(1 +2*x)*(1 +x^2)).
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 211; 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, Dec 21 2016
STATUS
approved