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 A280610 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 294", based on the 5-celled von Neumann neighborhood. 4
 1, 3, 1, 7, 17, 39, 65, 135, 257, 519, 1025, 2055, 4097, 8199, 16385, 32775, 65537, 131079, 262145, 524295, 1048577, 2097159, 4194305, 8388615, 16777217, 33554439, 67108865, 134217735, 268435457, 536870919, 1073741825, 2147483655, 4294967297, 8589934599 (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 + x - 6*x^2 + 4*x^3 + 8*x^4 - 16*x^6)/((1 - x)*(1 + x)*(1 - 2*x)). - Ilya Gutkovskiy, Jan 06 2017 Conjectures from Colin Barker, Jan 07 2017: (Start) a(n) = 4 - 3*(-1)^n + 2^n for n>3. a(n) = 2*a(n-1) + a(n-2) - 2*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[i, 2 * i - 1]], 2], {i , 1, stages - 1}] CROSSREFS Cf. A280606, A280607, A280608. Sequence in context: A101845 A193606 A221328 * A280416 A281738 A281109 Adjacent sequences: A280607 A280608 A280609 * A280611 A280612 A280613 KEYWORD nonn,easy AUTHOR Robert Price, Jan 06 2017 STATUS approved

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Last modified September 18 03:51 EDT 2024. Contains 375995 sequences. (Running on oeis4.)