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 A279053 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 131", based on the 5-celled von Neumann neighborhood. 5
 1, 1, 101, 101, 10101, 10101, 1010101, 1010101, 101010101, 101010101, 10101010101, 10101010101, 1010101010101, 1010101010101, 101010101010101, 101010101010101, 10101010101010101, 10101010101010101, 1010101010101010101, 1010101010101010101 (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 05 2016: (Start) a(n) = (9*(-2)^n*5^(n+1) + 11*2^n*5^(n+1) - 1)/99. a(n) = a(n-1) + 100*a(n-2) - 100*a(n-3) for n>2. G.f.: 1 / ((1 - x)*(1 - 10*x)*(1 + 10*x)). (End) MATHEMATICA CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code = 131; 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 Sequence in context: A283223 A279808 A279748 * A280145 A279528 A283044 Adjacent sequences:  A279050 A279051 A279052 * A279054 A279055 A279056 KEYWORD nonn,easy AUTHOR Robert Price, Dec 04 2016 STATUS approved

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Last modified December 15 14:37 EST 2019. Contains 329999 sequences. (Running on oeis4.)