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 A283605 Binary 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 670", based on the 5-celled von Neumann neighborhood. 4
 1, 11, 111, 1111, 10111, 111111, 1110111, 11111111, 101110111, 1111111111, 11011110111, 111111111111, 1100111110111, 11111111111111, 110111111110111, 1111111111111111, 11111111111110111, 111111111111111111, 1111111111111110111, 11111111111111111111 (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 Conjectures from Chai Wah Wu, May 06 2024: (Start) a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3) for n > 17. G.f.: (-10000000000000*x^17 + 1000000000000*x^16 + 9890000000000*x^15 - 989000000000*x^14 + 109000000000*x^13 - 10900000000*x^12 + 900000000*x^11 - 90000000*x^10 + 100000000*x^9 - 10000000*x^8 + 10000*x^5 - 1000*x^4 + x + 1)/((x - 1)*(x + 1)*(10*x - 1)). (End) MATHEMATICA CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code = 670; 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]], 10], {i, 1, stages - 1}] CROSSREFS Cf. A283604, A283606, A283607. Sequence in context: A284088 A284420 A283850 * A083440 A138144 A284243 Adjacent sequences: A283602 A283603 A283604 * A283606 A283607 A283608 KEYWORD nonn,easy AUTHOR Robert Price, Mar 11 2017 STATUS approved

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Last modified June 15 13:08 EDT 2024. Contains 373407 sequences. (Running on oeis4.)