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A281749
Binary 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 395", based on the 5-celled von Neumann neighborhood.
4
1, 10, 111, 10, 10111, 111010, 10111, 10101010, 111010111, 10111010, 10101010111, 111011101010, 10101010111, 10101110111010, 111010101010111, 10111011101010, 10101010101010111, 111011101010111010, 10101010101010111, 10101110101011101010
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) = - 10*a(n-1) + 1000*a(n-3) + 10000*a(n-4) + a(n-8) + 10*a(n-9) - 1000*a(n-11) - 10000*a(n-12) for n > 19.
G.f.: (-10000000000*x^19 - 1000000000*x^18 - 100000000*x^17 + 1000000*x^15 - 90000*x^12 + 991000*x^11 + 1100*x^9 - 99790*x^8 - 8880*x^7 + 211*x^6 + 1120*x^5 - 9789*x^4 + 120*x^3 + 211*x^2 + 20*x + 1)/(10000*x^12 + 1000*x^11 - 10*x^9 - x^8 - 10000*x^4 - 1000*x^3 + 10*x + 1). (End)
MATHEMATICA
CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 395; 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]], 10], {i, 1, stages - 1}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert Price, Jan 29 2017
STATUS
approved