|
|
A278666
|
|
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 62", based on the 5-celled von Neumann neighborhood.
|
|
4
|
|
|
1, 3, 6, 12, 24, 50, 102, 196, 388, 804, 1652, 3156, 6228, 12884, 26452, 50516, 99668, 206164, 423252, 808276, 1594708, 3298644, 6772052, 12932436, 25515348, 52778324, 108352852, 206918996, 408245588, 844453204, 1733645652, 3310703956, 6531929428
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = a(n-1) + 16*a(n-4) - 16*a(n-5) for n>7.
G.f.: (1 +2*x +3*x^2 +6*x^3 -4*x^4 -6*x^5 +4*x^6 -2*x^7 +16*x^10) / ((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=62; 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
|
|
|
STATUS
|
approved
|
|
|
|