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 A280608 Decimal 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 294", based on the 5-celled von Neumann neighborhood. 4
 1, 3, 4, 14, 17, 57, 65, 225, 257, 897, 1025, 3585, 4097, 14337, 16385, 57345, 65537, 229377, 262145, 917505, 1048577, 3670017, 4194305, 14680065, 16777217, 58720257, 67108865, 234881025, 268435457, 939524097, 1073741825, 3758096385, 4294967297, 15032385537 (list; graph; refs; listen; history; text; internal format)
 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. 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 Empirical g.f.: (1 + 2*x - 3*x^2 + 2*x^3 - x^4 - 4*x^6)/((1 - x)*(1 - 2*x)*(1 + 2*x)). - Ilya Gutkovskiy, Jan 06 2017 Conjectures from Colin Barker, Jan 07 2017: (Start) a(n) = (8 - 3*(-2)^n + 11*2^n) / 8 for n>3. a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n>4. (End) MATHEMATICA CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code = 294; 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]], 2], {i, 1, stages - 1}] CROSSREFS Cf. A280606, A280607, A280610. Sequence in context: A076663 A277933 A278953 * A280415 A281737 A281108 Adjacent sequences: A280605 A280606 A280607 * A280609 A280610 A280611 KEYWORD nonn,easy AUTHOR Robert Price, Jan 06 2017 STATUS approved

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Last modified September 16 00:43 EDT 2024. Contains 375959 sequences. (Running on oeis4.)