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 A273539 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 779", based on the 5-celled von Neumann neighborhood. 0
 1, 5, 37, 197, 901, 3845, 15877, 64517, 260101, 1044485, 4186117, 16760837, 67076101, 268369925, 1073610757, 4294705157 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Initialized with a single black (ON) cell at stage zero. Conjecture: Rules 795, 843 and 859 also generate this sequence. - Lars Blomberg, Jul 20 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 Conjectures from Colin Barker, May 25 2016: (Start) a(n) = 5-2^(3+n)+4^(1+n). a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>2. G.f.: (1-2*x+16*x^2) / ((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=779; 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. A273538. Sequence in context: A269815 A270931 A241629 * A095924 A270019 A179567 Adjacent sequences:  A273536 A273537 A273538 * A273540 A273541 A273542 KEYWORD nonn,more AUTHOR Robert Price, May 24 2016 EXTENSIONS a(8)-a(15) from Lars Blomberg, Jul 20 2016 STATUS approved

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Last modified April 23 03:51 EDT 2021. Contains 343199 sequences. (Running on oeis4.)