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 A290232 Binary representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 773", based on the 5-celled von Neumann neighborhood. 4
 1, 10, 111, 1100, 11111, 111000, 1111111, 11110000, 111111111, 1111100000, 11111111111, 111111000000, 1111111111111, 11111110000000, 111111111111111, 1111111100000000, 11111111111111111, 111111111000000000, 1111111111111111111, 11111111110000000000 (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 FORMULA Conjectures from Colin Barker, Jul 25 2017: (Start) G.f.: (1 - 10*x^3) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 - 10*x^2)). a(n) = (10^(n+1) - 1)/9 for n even. a(n) = (10^(n+1) - 10^((n+1)/2))/9 for n odd. a(n) = 10*a(n-1) + 11*a(n-2) - 110*a(n-3) - 10*a(n-4) + 100*a(n-5) 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 = 773; 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. A290233, A290234, A290235. Sequence in context: A105991 A284352 A103581 * A286855 A288210 A286823 Adjacent sequences:  A290229 A290230 A290231 * A290233 A290234 A290235 KEYWORD nonn,easy AUTHOR Robert Price, Jul 24 2017 STATUS approved

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Last modified September 20 02:20 EDT 2019. Contains 327207 sequences. (Running on oeis4.)