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A283905
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 758", based on the 5-celled von Neumann neighborhood.
4
1, 3, 5, 11, 21, 43, 93, 179, 381, 763, 1533, 3059, 6141, 12283, 24573, 49139, 98301, 196603, 393213, 786419, 1572861, 3145723, 6291453, 12582899, 25165821, 50331643, 100663293, 201326579, 402653181, 805306363, 1610612733, 3221225459, 6442450941, 12884901883
OFFSET
0,2
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, Dec 10 2017: (Start)
G.f.: (1 + x - x^2 + x^3 - 2*x^4 + 8*x^6 - 8*x^7 + 24*x^8) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + x^2)).
a(n) = 2*a(n-1) + a(n-4) - 2*a(n-5) for n>4.
(End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 758; 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, Mar 17 2017
STATUS
approved