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A281281
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 350", based on the 5-celled von Neumann neighborhood.
4
1, 11, 110, 1101, 11011, 110100, 1101110, 11010011, 110111100, 1101000010, 11011101101, 110100111111, 1101111000000, 11010000100010, 110111011010101, 1101001100101111, 11011110111011000, 110100001000100110, 1101110110101011001, 11010011001010100111
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 05 2024: (Start)
a(n) = 10001*a(n-4) - 10000*a(n-8) for n > 23.
G.f.: (-990000000*x^23 + 991000000*x^22 - 1000000*x^21 - 110000000*x^20 + 9999000*x^19 + 900*x^18 + 100*x^17 + 111011000*x^16 - 11010000*x^15 - 1000*x^14 - 1100*x^12 + 1100*x^11 - 9*x^10 - 90*x^9 + 89*x^8 - 1090*x^7 + 1000*x^6 + 89*x^5 + 1010*x^4 + 1101*x^3 + 110*x^2 + 11*x + 1)/(10000*x^8 - 10001*x^4 + 1). (End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 350; 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, Jan 18 2017
STATUS
approved