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 A273570 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 798", based on the 5-celled von Neumann neighborhood. 0
 1, 5, 21, 93, 429, 1869, 7821, 32013, 129549, 521229, 2091021, 8376333, 33529869, 134168589, 536772621, 2147287053 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Initialized with a single black (ON) cell at stage zero. Conjecture: Rule 862 also generates this sequence. - Lars Blomberg, Jul 21 2016 REFERENCES S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170. LINKS 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 FORMULA Conjecture: a(n) = 2*4^n - 6*2^n + 13, n>1. - Lars Blomberg, Jul 21 2016 Conjectures from Colin Barker, Dec 01 2016: (Start) a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>4. G.f.: (1 - 2*x + 8*x^3 + 32*x^4) / ((1 - x)*(1 - 2*x)*(1 - 4*x)). (End) MATHEMATICA CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code=798; 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}]; on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *) Part[on, 2^Range[0, Log[2, stages]]] (* Extract relevant terms *) CROSSREFS Cf. A273569. Sequence in context: A007287 A116904 A126952 * A103519 A178876 A202513 Adjacent sequences:  A273567 A273568 A273569 * A273571 A273572 A273573 KEYWORD nonn,more AUTHOR Robert Price, May 25 2016 EXTENSIONS a(8)-a(15) from Lars Blomberg, Jul 21 2016 STATUS approved

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Last modified February 23 04:24 EST 2019. Contains 320411 sequences. (Running on oeis4.)