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 A271052 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 253", based on the 5-celled von Neumann neighborhood. 1
 1, 9, 13, 57, 70, 186, 199, 419, 432, 788, 801, 1325, 1338, 2062, 2075, 3031, 3044, 4264, 4277, 5793, 5806, 7650, 7663, 9867, 9880, 12476, 12489, 15509, 15522, 18998, 19011, 22975, 22988, 27472, 27485, 32521, 32534, 38154, 38167, 44403, 44416, 51300, 51313 (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 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, Nov 22 2017: (Start) G.f.: (1 + 8*x + x^2 + 20*x^3 + 4*x^4 + 8*x^5 - 15*x^6 - 4*x^7 + 9*x^8) / ((1 - x)^4*(1 + x)^3). a(n) = (8*n^3 + 12*n^2 + 46*n - 48) / 12 for n>1 and even. a(n) = (8*n^3 + 36*n^2 + 94*n - 138) / 12 for n>1 and odd. a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>8. (End) MATHEMATICA CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code=253; 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. A271051. Sequence in context: A270936 A365239 A271006 * A305650 A099580 A258286 Adjacent sequences: A271049 A271050 A271051 * A271053 A271054 A271055 KEYWORD nonn,easy AUTHOR Robert Price, Mar 29 2016 STATUS approved

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Last modified September 12 05:07 EDT 2024. Contains 375842 sequences. (Running on oeis4.)