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 A286771 Binary representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 221", based on the 5-celled von Neumann neighborhood. 4
 1, 1, 0, 111, 10000, 11111, 1000000, 1111111, 100000000, 111111111, 10000000000, 11111111111, 1000000000000, 1111111111111, 100000000000000, 111111111111111, 10000000000000000, 11111111111111111, 1000000000000000000, 1111111111111111111 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Initialized with a single black (ON) cell at stage zero. REFERENCES S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170. LINKS Robert Price, Table of n, a(n) for n = 0..126 Robert Price, Diagrams of first 20 stages N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015 Eric Weisstein's World of Mathematics, Elementary Cellular Automaton S. Wolfram, A New Kind of Science Wolfram Research, Wolfram Atlas of Simple Programs FORMULA Conjectures from Colin Barker, May 14 2017: (Start) G.f.: (1 + x - 101*x^2 + 10*x^3 + 10100*x^4 - 10000*x^6) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)). a(n) = 10^n for n>2 and even. a(n) = (10^n - 1)/9 for n>2 and odd. a(n) = 101*a(n-2) - 100*a(n-4) for n>4. (End) It appears that a(n) = A280411(n) for n >= 3. - Michel Marcus, May 20 2017 MATHEMATICA CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code = 221; 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 Cf. A286770, A286772, A286773. Sequence in context: A286117 A286083 A286731 * A284208 A266381 A118109 Adjacent sequences:  A286768 A286769 A286770 * A286772 A286773 A286774 KEYWORD nonn,easy AUTHOR Robert Price, May 14 2017 STATUS approved

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Last modified June 19 05:20 EDT 2019. Contains 324217 sequences. (Running on oeis4.)