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 A286773 Decimal representation of the diagonal from the origin to the corner 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, 0, 7, 16, 31, 64, 127, 256, 511, 1024, 2047, 4096, 8191, 16384, 32767, 65536, 131071, 262144, 524287, 1048576, 2097151, 4194304, 8388607, 16777216, 33554431, 67108864, 134217727, 268435456, 536870911, 1073741824, 2147483647, 4294967296, 8589934591 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 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, May 15 2017: (Start) G.f.: (1 - x - 3*x^2 + 8*x^3 + 4*x^4 - 8*x^5) / ((1 - x)*(1 + x)*(1 - 2*x)). a(n) = 2^n for n>2 and even. a(n) = 2^n - 1 for n>2 and odd. a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>5. (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]], 10], {i, 1, stages - 1}] CROSSREFS Cf. A286770, A286771, A286772. Sequence in context: A286085 A121470 A286733 * A019541 A101426 A296153 Adjacent sequences:  A286770 A286771 A286772 * A286774 A286775 A286776 KEYWORD nonn,easy AUTHOR Robert Price, May 14 2017 STATUS approved

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Last modified December 13 03:41 EST 2019. Contains 329968 sequences. (Running on oeis4.)