login
A282577
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 553", based on the 5-celled von Neumann neighborhood.
5
1, 0, 1, 0, 5, 0, 21, 0, 85, 0, 341, 0, 1365, 0, 5461, 0, 21845, 0, 87381, 0, 349525, 0, 1398101, 0, 5592405, 0, 22369621, 0, 89478485, 0, 357913941, 0, 1431655765, 0, 5726623061, 0, 22906492245, 0, 91625968981, 0, 366503875925, 0, 1466015503701, 0
OFFSET
0,5
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, Feb 27 2017: (Start)
a(n) = (2^n - 1) / 3 for n>1 and even.
a(n) = 0 for n>1 and odd.
a(n) = 5*a(n-2) - 4*a(n-4) for n>4.
G.f.: (1 - 2*x^2)^2 / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 553; 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
Robert Price, Feb 27 2017
STATUS
approved