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A283606
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 670", based on the 5-celled von Neumann neighborhood.
4
1, 3, 7, 15, 29, 63, 119, 255, 477, 1023, 1915, 4095, 7667, 16383, 30715, 65535, 122879, 262143, 491519, 1048575, 1966079, 4194303, 7864319, 16777215, 31457279, 67108863, 125829119, 268435455, 503316479, 1073741823, 2013265919, 4294967295, 8053063679
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 Chai Wah Wu, May 06 2024: (Start)
a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n > 17.
G.f.: (-16*x^17 + 16*x^16 - 44*x^15 + 44*x^14 - 4*x^13 + 4*x^12 - 4*x^11 + 4*x^10 + 2*x^9 - 2*x^8 + 2*x^5 - 2*x^4 + 2*x + 1)/((x - 1)*(2*x - 1)*(2*x + 1)). (End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 670; 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 11 2017
STATUS
approved