|
| |
|
|
A277799
|
|
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 1", based on the 5-celled von Neumann neighborhood.
|
|
8
|
|
|
|
1, 0, 1, 12, 1, 60, 1, 252, 1, 1020, 1, 4092, 1, 16380, 1, 65532, 1, 262140, 1, 1048572, 1, 4194300, 1, 16777212, 1, 67108860, 1, 268435452, 1, 1073741820, 1, 4294967292, 1, 17179869180, 1, 68719476732, 1, 274877906940, 1, 1099511627772, 1, 4398046511100, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,4
|
|
|
COMMENTS
|
Initialized with a single black (ON) cell at stage zero.
Rule numbers 1, 9, 17, 25, 257, 265, 273 and 281 all generate this sequence.
|
|
|
REFERENCES
|
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
|
|
|
LINKS
|
|
|
|
FORMULA
|
G.f.: (1 - 4*x^2 + 12*x^3) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
a(n) = (-3/2-(-2)^n+(5*(-1)^n)/2+2^n). (End)
|
|
|
MATHEMATICA
|
CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=1; 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
|
| |
|
|
