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 A269782 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 65", based on the 5-celled von Neumann neighborhood. 5
 1, 4, 5, 36, 9, 96, 17, 188, 21, 312, 25, 468, 29, 656, 33, 876, 37, 1128, 41, 1412, 45, 1728, 49, 2076, 53, 2456, 57, 2868, 61, 3312, 65, 3788, 69, 4296, 73, 4836, 77, 5408, 81, 6012, 85, 6648, 89, 7316, 93, 8016, 97, 8748, 101, 9512, 105, 10308, 109, 11136 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Initialized with a single black (ON) cell at stage zero. Similar to A270569. REFERENCES S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170. LINKS Robert Price, Table of n, a(n) for n = 0..128 Robert Price, Diagrams of the first 20 stages. 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, Apr 03 2016: (Start) a(n) = (11-(-1)^n+4*(-1)^n*n-4*(-1+(-1)^n)*n^2)/2 for n>4. a(n) = 2*n+5 for n>4 and even. a(n) = 4*n^2-2*n+6 for n>4 and odd. a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>8. G.f.: (1+4*x+2*x^2+24*x^3-3*x^4+4*x^6+4*x^7-8*x^8+4*x^10) / ((1-x)^3*(1+x)^3). (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}]; Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *) CROSSREFS Cf. A270569. Sequence in context: A243772 A332316 A243275 * A131139 A152291 A336024 Adjacent sequences:  A269779 A269780 A269781 * A269783 A269784 A269785 KEYWORD nonn,easy AUTHOR Robert Price, Mar 10 2016 STATUS approved

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Last modified April 20 12:36 EDT 2021. Contains 343135 sequences. (Running on oeis4.)