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 A273316 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 645", based on the 5-celled von Neumann neighborhood. 1
 1, 9, 33, 81, 161, 281, 449, 673, 961, 1321, 1761, 2289, 2913, 3641, 4481, 5441, 6529, 7753, 9121, 10641, 12321, 14169, 16193, 18401, 20801, 23401, 26209, 29233, 32481, 35961, 39681, 43649, 47873, 52361, 57121, 62161, 67489, 73113, 79041, 85281, 91841, 98729 (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+8*n+3)/3. a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>3. G.f.: (1+x)*(1+4*x-x^2) / (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=645; 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 *) Table[Total[Part[on, Range[1, i]]], {i, 1, Length[on]}] (* Sum at each stage *) CROSSREFS Cf. A033996. Sequence in context: A205796 A081585 A227221 * A101990 A147170 A146823 Adjacent sequences:  A273313 A273314 A273315 * A273317 A273318 A273319 KEYWORD nonn,easy AUTHOR Robert Price, May 19 2016 STATUS approved

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Last modified June 16 02:14 EDT 2021. Contains 345055 sequences. (Running on oeis4.)