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 A283591 Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 657", based on the 5-celled von Neumann neighborhood. 4
 1, 0, 5, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534, 131070, 262142, 524286, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 67108862, 134217726, 268435454, 536870910, 1073741822, 2147483646, 4294967294, 8589934590 (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, Mar 13 2017: (Start) G.f.: (1 - 3*x + 7*x^2 - x^3 - 2*x^4) / ((1 - x)*(1 - 2*x)). a(n) = 2*(2^n - 1) for n>2. a(n) = 3*a(n-1) - 2*a(n-2) 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 = 657; 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[1, i]], 2], {i, 1, stages - 1}] CROSSREFS Cf. A283589, A283590, A283592. Sequence in context: A299902 A299899 A211804 * A238604 A166068 A070129 Adjacent sequences:  A283588 A283589 A283590 * A283592 A283593 A283594 KEYWORD nonn,easy AUTHOR Robert Price, Mar 11 2017 STATUS approved

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Last modified May 18 16:07 EDT 2021. Contains 343995 sequences. (Running on oeis4.)