|
| |
|
|
A287470
|
|
Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 276", based on the 5-celled von Neumann neighborhood.
|
|
4
|
|
|
|
1, 0, 6, 0, 28, 0, 120, 0, 496, 0, 2016, 0, 8128, 0, 32640, 0, 130816, 0, 523776, 0, 2096128, 0, 8386560, 0, 33550336, 0, 134209536, 0, 536854528, 0, 2147450880, 0, 8589869056, 0, 34359607296, 0, 137438691328, 0, 549755289600, 0, 2199022206976, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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.
|
|
|
LINKS
|
|
|
|
FORMULA
|
Conjectures:
G.f.: 1 / ((1 - 2*x)*(1 + 2*x)*(1 - 2*x^2)).
a(n) = 2^(n+1) - 2^(n/2) for n even.
a(n) = 0 for n odd.
a(n) = 6*a(n-2) - 8*a(n-4) 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 = 276; 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]], 10], {i, 1, stages - 1}]
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
