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 A282799 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 505", based on the 5-celled von Neumann neighborhood. 4
 1, 0, 7, 2, 31, 10, 127, 42, 511, 170, 2047, 682, 8191, 2730, 32767, 10922, 131071, 43690, 524287, 174762, 2097151, 699050, 8388607, 2796202, 33554431, 11184810, 134217727, 44739242, 536870911, 178956970, 2147483647, 715827882, 8589934591, 2863311530 (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 Index entries for sequences related to cellular automata Index to 2D 5-Neighbor Cellular Automata Index to Elementary Cellular Automata FORMULA Conjectures from Colin Barker, Feb 21 2017: (Start) a(n) = 2^(n+1) - 1 for n even. a(n) = (2^n-2) / 3 for n odd. a(n) = 5*a(n-2) - 4*a(n-4) for n>3. G.f.: (1 + 2*x^2 + 2*x^3) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)). (End) MATHEMATICA CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code = 505; 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]], 2], {i , 1, stages - 1}] CROSSREFS Cf. A282796, A282797, A282798. Sequence in context: A280337 A195587 A096900 * A222555 A246799 A248189 Adjacent sequences: A282796 A282797 A282798 * A282800 A282801 A282802 KEYWORD nonn,easy AUTHOR Robert Price, Feb 21 2017 STATUS approved

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Last modified April 18 10:01 EDT 2024. Contains 371779 sequences. (Running on oeis4.)