|
| |
|
|
A277797
|
|
Binary 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.
|
|
4
|
|
|
|
1, 0, 1, 1100, 1, 111100, 1, 11111100, 1, 1111111100, 1, 111111111100, 1, 11111111111100, 1, 1111111111111100, 1, 111111111111111100, 1, 11111111111111111100, 1, 1111111111111111111100, 1, 111111111111111111111100, 1, 11111111111111111111111100, 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 - 100*x^2 + 1100*x^3)/((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).
a(n) = 101*a(n-2) - 100*a(n-4) for n>3.
a(n) = (-91+109*(-1)^n+10^(1+n)-(-1)^n*10^(1+n))/18. (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]], 10], {i, 1, stages-1}]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
