login
A277867
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 3", based on the 5-celled von Neumann neighborhood.
4
1, 2, 4, 11, 16, 47, 64, 191, 256, 767, 1024, 3071, 4096, 12287, 16384, 49151, 65536, 196607, 262144, 786431, 1048576, 3145727, 4194304, 12582911, 16777216, 50331647, 67108864, 201326591, 268435456, 805306367, 1073741824, 3221225471, 4294967296, 12884901887
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, Nov 03 2016: (Start)
G.f.: (1+2*x-x^2+x^3) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)).
a(n) = (-2-(-2)^n+2*(-1)^n+5*2^n)/4. (End)
MATHEMATICA
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=3; 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, Nov 02 2016
STATUS
approved