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A282799
Decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 505", based on the 5-celled von Neumann neighborhood.
4
1, 0, 7, 2, 31, 10, 127, 42, 511, 170, 2047, 682, 8191, 2730, 32767, 10922, 131071, 43690, 524287, 174762, 2097151, 699050, 8388607, 2796202, 33554431, 11184810, 134217727, 44739242, 536870911, 178956970, 2147483647, 715827882, 8589934591, 2863311530
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.
FORMULA
Conjectures from Colin Barker, Feb 21 2017: (Start)
a(n) = 2^(n+1) - 1 for n even.
a(n) = (2^n-2) / 3 for n odd.
a(n) = 5*a(n-2) - 4*a(n-4) for n>3.
G.f.: (1 + 2*x^2 + 2*x^3) / ((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 = 505; 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]], 2], {i , 1, stages - 1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Feb 21 2017
STATUS
approved