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 A270222 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 131", based on the 5-celled von Neumann neighborhood. 1
 1, 5, 33, 161, 705, 2945, 12033, 48641, 195585, 784385, 3141633, 12574721, 50315265, 201293825, 805240833, 3221094401 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Initialized with a single black (ON) cell at stage zero. It is conjectured that Rules 163, 171, 227, 235, 771, 787, 803, 811, 819, 827, 835, 851, 867, 875, 883 and 891 also generate this sequence. - Lars Blomberg, Apr 30 2016 Also the number of vertex cuts in the (n+1)-barbell graph for n > 1. - Eric W. Weisstein, Apr 23 2023 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, Barbell Graph Eric Weisstein's World of Mathematics, Elementary Cellular Automaton Eric Weisstein's World of Mathematics, Vertex Cut 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, Mar 13 2016: (Start) a(n) = 1-2^(2+n)+3*4^n. a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3. G.f.: (1-2*x+12*x^2-8*x^3) / ((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=131; 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)/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. A270221. Sequence in context: A146263 A359703 A255943 * A270279 A270690 A273141 Adjacent sequences: A270219 A270220 A270221 * A270223 A270224 A270225 KEYWORD nonn,more AUTHOR Robert Price, Mar 13 2016 EXTENSIONS a(8)-a(15) from Lars Blomberg, Apr 30 2016 STATUS approved

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Last modified June 7 22:39 EDT 2023. Contains 363157 sequences. (Running on oeis4.)