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 A270084 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 65", based on the 5-celled von Neumann neighborhood. 1
 1, 4, 36, 188, 876, 3788, 15756, 64268, 259596, 1043468, 4184076, 16756748, 67067916, 268353548, 1073577996, 4294639628 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Initialized with a single black (ON) cell at stage zero. Similar to A271254. Lars Blomberg conjectured that Rule 321 also produces this sequence.  It would be nice to have a proof. 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) = 4*4^n - 10*2^n + 12, n>1. - Lars Blomberg, Apr 21 2016 Conjectures from Colin Barker, Apr 21 2016: (Start) a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>4. G.f.: (1-3*x+22*x^2-16*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=65; 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. A269782, A271254. Sequence in context: A276295 A272146 A034569 * A272222 A263420 A302946 Adjacent sequences:  A270081 A270082 A270083 * A270085 A270086 A270087 KEYWORD nonn,more AUTHOR Robert Price, Mar 10 2016 EXTENSIONS a(8)-a(15) from Lars Blomberg, Apr 21 2016 STATUS approved

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Last modified April 9 14:02 EDT 2020. Contains 333353 sequences. (Running on oeis4.)