login
A283507
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 641", based on the 5-celled von Neumann neighborhood.
6
1, 0, 5, 3, 23, 15, 95, 63, 383, 255, 1535, 1023, 6143, 4095, 24575, 16383, 98303, 65535, 393215, 262143, 1572863, 1048575, 6291455, 4194303, 25165823, 16777215, 100663295, 67108863, 402653183, 268435455, 1610612735, 1073741823, 6442450943, 4294967295
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 10 2017: (Start)
G.f.: (1 - x + x^2 + 2*x^3) / ((1 - x)*(1 - 2*x)*(1 + 2*x)).
a(n) = (-2 + (-2)^n + 2^(1+n)) / 2 for n>0.
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n>3.
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 641; 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 09 2017
STATUS
approved