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 A283358 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 621", based on the 5-celled von Neumann neighborhood. 4
 1, 1, 1, 13, 17, 29, 49, 125, 241, 509, 1009, 2045, 4081, 8189, 16369, 32765, 65521, 131069, 262129, 524285, 1048561, 2097149, 4194289, 8388605, 16777201, 33554429, 67108849, 134217725, 268435441, 536870909, 1073741809, 2147483645, 4294967281, 8589934589 (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, Mar 07 2017: (Start) G.f.: (1 - x - 2*x^2 + 12*x^3 - 8*x^4 - 16*x^5 + 32*x^7) / ((1 - x)*(1 + x)*(1 - 2*x)). a(n) = 2^n - 15 for n>4 and even. a(n) = 2^n - 3 for n>4 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 = 621; 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. A283355, A283356, A283357. Sequence in context: A061381 A048520 A283407 * A033210 A274506 A107159 Adjacent sequences:  A283355 A283356 A283357 * A283359 A283360 A283361 KEYWORD nonn,easy AUTHOR Robert Price, Mar 05 2017 STATUS approved

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Last modified September 30 18:12 EDT 2020. Contains 337440 sequences. (Running on oeis4.)