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 A273335 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 657", based on the 5-celled von Neumann neighborhood. 0
 1, 4, 48, 224, 960, 3968, 16128, 65024, 261120, 1046528, 4190208, 16769024, 67092480, 268402688, 1073676288, 4294836224 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Initialized with a single black (ON) cell at stage zero. Conjecture: Rule 665 also generates this sequence. - Lars Blomberg, Jul 18 2016 REFERENCES S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170. LINKS Table of n, a(n) for n=0..15. 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 Index entries for sequences related to cellular automata Index to 2D 5-Neighbor Cellular Automata Index to Elementary Cellular Automata FORMULA Conjectures from Colin Barker, May 20 2016: (Start) a(n) = 2^(n+2)*(2^n-1) for n>1. a(n) = 6*a(n-1)-8*a(n-2) for n>3. G.f.: (1-2*x+32*x^2-32*x^3) / ((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=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}]; 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. A273334. Sequence in context: A192418 A162673 A002287 * A275138 A273767 A292074 Adjacent sequences: A273332 A273333 A273334 * A273336 A273337 A273338 KEYWORD nonn,more AUTHOR Robert Price, May 20 2016 EXTENSIONS a(8)-a(15) from Lars Blomberg, Jul 18 2016 STATUS approved

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Last modified August 15 12:15 EDT 2024. Contains 375173 sequences. (Running on oeis4.)