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 A283584 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 646", based on the 5-celled von Neumann neighborhood. 4
 1, 3, 5, 11, 21, 43, 69, 219, 261, 795, 1301, 2923, 5205, 11691, 20805, 46811, 83205, 187163, 333077, 748395, 1332309, 2993579, 5329221, 11974363, 21316869, 47897371, 85267733, 191589227, 341070933, 766356907, 1364283717, 3065427675, 5457134853, 12261710619 (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, Mar 12 2017: (Start) G.f.: (1 + 3*x + x^2 - x^3 + x^4 - x^5 - 15*x^6 + 47*x^7 - 16*x^8 - 84*x^9 + 256*x^10 - 256*x^11) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 + x^2)*(1 + x^4)). a(n) = 4*a(n-2) + a(n-8) - 4*a(n-10) for n>11. (End) MATHEMATICA CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code = 646; 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]], 2], {i , 1, stages - 1}] CROSSREFS Cf. A283581, A283582, A283583. Sequence in context: A146042 A291219 A284358 * A283702 A284539 A154917 Adjacent sequences:  A283581 A283582 A283583 * A283585 A283586 A283587 KEYWORD nonn,easy AUTHOR Robert Price, Mar 11 2017 STATUS approved

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Last modified October 27 21:22 EDT 2020. Contains 338036 sequences. (Running on oeis4.)