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 A273314 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood. 1
 1, 6, 23, 64, 137, 250, 411, 628, 909, 1262, 1695, 2216, 2833, 3554, 4387, 5340, 6421, 7638, 8999, 10512, 12185, 14026, 16043, 18244, 20637, 23230, 26031, 29048, 32289, 35762, 39475, 43436, 47653, 52134, 56887, 61920, 67241, 72858, 78779, 85012, 91565, 98446 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Initialized with a single black (ON) cell at stage zero. 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 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 19 2016: (Start) a(n) = (4*n^3+12*n^2-13*n+15)/3 for n>0. a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4. G.f.: (1+2*x+5*x^2+4*x^3-4*x^4) / (1-x)^4. (End) MATHEMATICA CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code=643; 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 *) Table[Total[Part[on, Range[1, i]]], {i, 1, Length[on]}] (* Sum at each stage *) CROSSREFS Cf. A166147. Sequence in context: A273252 A208598 A119712 * A281424 A005745 A213557 Adjacent sequences:  A273311 A273312 A273313 * A273315 A273316 A273317 KEYWORD nonn,easy AUTHOR Robert Price, May 19 2016 STATUS approved

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Last modified December 17 14:12 EST 2018. Contains 318201 sequences. (Running on oeis4.)