login
A279875
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 211", based on the 5-celled von Neumann neighborhood.
4
1, 1, 5, 7, 16, 23, 84, 125, 257, 383, 1344, 2015, 4096, 6143, 21504, 32255, 65536, 98303, 344064, 516095, 1048576, 1572863, 5505024, 8257535, 16777216, 25165823, 88080384, 132120575, 268435456, 402653183, 1409286144, 2113929215, 4294967296, 6442450943
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, Dec 21 2016: (Start)
a(n) = a(n-2) + 16*a(n-4) - 16*a(n-6) for n>10.
G.f.: (1 +x +4*x^2 +6*x^3 -5*x^4 +4*x^6 +6*x^7 -3*x^8 +2*x^9 -x^10 -16*x^12 +16*x^14) / ((1 -x)*(1 +x)*(1 -2*x)*(1 +2*x)*(1 +4*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[1, i]], 2], {i, 1, stages - 1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Dec 21 2016
STATUS
approved