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 A285777 Decimal 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 89", based on the 5-celled von Neumann neighborhood. 4
 1, 2, 1, 14, 0, 63, 0, 255, 0, 1023, 0, 4095, 0, 16383, 0, 65535, 0, 262143, 0, 1048575, 0, 4194303, 0, 16777215, 0, 67108863, 0, 268435455, 0, 1073741823, 0, 4294967295, 0, 17179869183, 0, 68719476735, 0, 274877906943, 0, 1099511627775, 0, 4398046511103, 0 (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, Apr 26 2017: (Start) G.f.: (1 + 2*x - 4*x^2 + 4*x^3 - x^4 + x^5 + 4*x^6 - 4*x^7) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)). a(n) = 0 for n>3 and even. a(n) = 2^(n+1) - 1 for n>3 and odd. a(n) = 5*a(n-2) - 4*a(n-4) for n>7. (End) MATHEMATICA CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code = 89; 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)/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. A285775, A285776, A285778. Sequence in context: A013017 A012902 A087169 * A286700 A286740 A285653 Adjacent sequences:  A285774 A285775 A285776 * A285778 A285779 A285780 KEYWORD nonn,easy AUTHOR Robert Price, Apr 25 2017 STATUS approved

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Last modified October 14 09:25 EDT 2019. Contains 327995 sequences. (Running on oeis4.)