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 A279962 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 221", based on the 5-celled von Neumann neighborhood. 4
 1, 1, 2, 3, 28, 15, 112, 63, 448, 255, 1792, 1023, 7168, 4095, 28672, 16383, 114688, 65535, 458752, 262143, 1835008, 1048575, 7340032, 4194303, 29360128, 16777215, 117440512, 67108863, 469762048, 268435455, 1879048192, 1073741823, 7516192768, 4294967295 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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, Dec 24 2016: (Start) a(n) = 2^(n-3) * (5*(-1)^n + 9) for n>2 and even. a(n) = (5*(-2)^n + 9*2^n - 8)/8 for n>2 and odd. a(n) = 5*a(n-2) - 4*a(n-4) for n>6. G.f.: (1 + x - 3*x^2 - 2*x^3 + 22*x^4 + 4*x^5 - 20*x^6) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)). (End) MATHEMATICA CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code = 221; 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. A279959, A279960, A279961. Sequence in context: A279755 A279148 A279825 * A037320 A010344 A037316 Adjacent sequences:  A279959 A279960 A279961 * A279963 A279964 A279965 KEYWORD nonn,easy AUTHOR Robert Price, Dec 23 2016 STATUS approved

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Last modified September 21 12:54 EDT 2021. Contains 347598 sequences. (Running on oeis4.)