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A284353
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 899", based on the 5-celled von Neumann neighborhood.
4
1, 1, 7, 13, 31, 61, 127, 253, 511, 1021, 2047, 4093, 8191, 16381, 32767, 65533, 131071, 262141, 524287, 1048573, 2097151, 4194301, 8388607, 16777213, 33554431, 67108861, 134217727, 268435453, 536870911, 1073741821, 2147483647, 4294967293, 8589934591
OFFSET
0,3
COMMENTS
Initialized with a single black (ON) cell at stage zero.
If one begins the Generalized Jacobsthal numbers (A083579) with a(0)=1, instead of a(0)=0, the same sequence will be obtained. - Henrik Lipskoch, Jan 28 2021
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
FORMULA
Conjectures from Colin Barker, Mar 26 2017: (Start)
G.f.: (1 - x + 4*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
a(n) = 2^(n+1) - 1 for n even.
a(n) = 2^(n+1) - 3 for n odd.
a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. (End)
Conjecture: For n > 0, a(n) = Sum_{k=0..n-1} C(n,k) * (2-(-1)^k). - Wesley Ivan Hurt, Sep 23 2017
Apparently, a(n) = 6*A000975(n-1) + 1 for n >= 1. - Hugo Pfoertner, Jan 28 2021
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 899; 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, Mar 25 2017
STATUS
approved