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A283849
Binary 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 734", based on the 5-celled von Neumann neighborhood.
4
1, 11, 111, 1111, 11101, 111111, 1110111, 11111111, 111011101, 1111111111, 11101111011, 111111111011, 1110111011111, 11111111111111, 111011110110111, 1111111111111111, 11101110111111111, 111111111111111111, 1110111101101111111, 11111111111111111111
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) + 10000*a(n-4) - 10000*a(n-5) for n > 25.
G.f.: (-1000000000000*x^25 + 1000000000000*x^24 - 100000000000*x^23 + 100000000000*x^22 + 100000000*x^21 - 100000000*x^20 - 1000000*x^16 + 1001000*x^15 - 1000*x^14 + 100*x^12 - 100*x^10 + 10*x^9 - 10*x^8 + 1000*x^7 - 1000*x^6 + 10*x^5 - 10*x^4 + 1000*x^3 + 100*x^2 + 10*x + 1)/(10000*x^5 - 10000*x^4 - x + 1). (End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 734; 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]], 10], {i, 1, stages - 1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Mar 17 2017
STATUS
approved