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 A279880 Decimal 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 213", based on the 5-celled von Neumann neighborhood. 4
 1, 1, 2, 3, 28, 7, 120, 15, 496, 31, 2016, 63, 8128, 127, 32640, 255, 130816, 511, 523776, 1023, 2096128, 2047, 8386560, 4095, 33550336, 8191, 134209536, 16383, 536854528, 32767, 2147450880, 65535, 8589869056, 131071, 34359607296, 262143, 137438691328 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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, Dec 22 2016: (Start) a(n) = 7*a(n-2)-14*a(n-4)+8*a(n-6) for n>6. G.f.: (1+x-5*x^2-4*x^3+28*x^4-56*x^6+32*x^8) / ((1-x)*(1+x)*(1-2*x)*(1+2*x)*(1-2*x^2)). (End) MATHEMATICA CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code = 213; 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)/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]], 2], {i , 1, stages - 1}] CROSSREFS Cf. A279877, A279878, A279879. Sequence in context: A098812 A073049 A349826 * A279140 A279755 A279148 Adjacent sequences: A279877 A279878 A279879 * A279881 A279882 A279883 KEYWORD nonn,easy AUTHOR Robert Price, Dec 21 2016 STATUS approved

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Last modified February 9 05:01 EST 2023. Contains 360153 sequences. (Running on oeis4.)