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 A270089 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 73", based on the 5-celled von Neumann neighborhood. 1
 1, 5, 10, 50, 50, 171, 171, 396, 396, 757, 757, 1286, 1286, 2015, 2015, 2976, 2976, 4201, 4201, 5722, 5722, 7571, 7571, 9780, 9780, 12381, 12381, 15406, 15406, 18887, 18887, 22856, 22856, 27345, 27345, 32386, 32386, 38011, 38011, 44252, 44252, 51141, 51141 (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, Mar 11 2016: (Start) a(n) = (-87-9*(-1)^n+(22-24*(-1)^n)*n-12*(-2+(-1)^n)*n^2+8*n^3)/12 for n>2. a(n) = (4*n^3+6*n^2-n-48)/6 for n>2 and even. a(n) = (4*n^3+18*n^2+23*n-39)/6 for n>2 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>9. G.f.: (1+4*x+2*x^2+28*x^3-12*x^4+13*x^5+14*x^6-22*x^7-5*x^8+9*x^9) / ((1-x)^4*(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=73; 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. A270087. Sequence in context: A034190 A216390 A305476 * A336582 A271275 A270100 Adjacent sequences:  A270086 A270087 A270088 * A270090 A270091 A270092 KEYWORD nonn,easy AUTHOR Robert Price, Mar 10 2016 STATUS approved

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Last modified May 9 04:52 EDT 2021. Contains 343687 sequences. (Running on oeis4.)