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 A271091 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 275", based on the 5-celled von Neumann neighborhood. 4
 1, 5, 5, 40, 5, 112, 5, 216, 5, 352, 5, 520, 5, 720, 5, 952, 5, 1216, 5, 1512, 5, 1840, 5, 2200, 5, 2592, 5, 3016, 5, 3472, 5, 3960, 5, 4480, 5, 5032, 5, 5616, 5, 6232, 5, 6880, 5, 7560, 5, 8272, 5, 9016, 5, 9792, 5, 10600, 5, 11440, 5, 12312, 5, 13216, 5 (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 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, Mar 30 2016: (Start) a(n) = (-3+13*(-1)^n-4*(-1+(-1)^n)*n-4*(-1+(-1)^n)*n^2)/2 for n>1. a(n) = 5 for n>1 and even. a(n) = 4*n^2+4*n-8 for n>1 and odd. a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>7. G.f.: (1+5*x+2*x^2+25*x^3-7*x^4+7*x^5+4*x^6-5*x^7) / ((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=275; 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}]; Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *) CROSSREFS Sequence in context: A139386 A074947 A099757 * A271289 A271083 A271277 Adjacent sequences:  A271088 A271089 A271090 * A271092 A271093 A271094 KEYWORD nonn,easy AUTHOR Robert Price, Mar 30 2016 STATUS approved

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Last modified September 30 10:16 EDT 2020. Contains 337438 sequences. (Running on oeis4.)