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 A271051 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. 3
 1, 8, 4, 44, 13, 116, 13, 220, 13, 356, 13, 524, 13, 724, 13, 956, 13, 1220, 13, 1516, 13, 1844, 13, 2204, 13, 2596, 13, 3020, 13, 3476, 13, 3964, 13, 4484, 13, 5036, 13, 5620, 13, 6236, 13, 6884, 13, 7564, 13, 8276, 13, 9020, 13, 9796, 13, 10604, 13, 11444 (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, 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)^3*(1 + x)^3). a(n) = 13 for n>2 and even. a(n) = 4*(n^2 + n - 1) for n>2 and odd. a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) 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)/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: A270901 A270934 A271004 * A046106 A112584 A112546 Adjacent sequences:  A271048 A271049 A271050 * A271052 A271053 A271054 KEYWORD nonn,easy AUTHOR Robert Price, Mar 29 2016 STATUS approved

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Last modified June 25 13:19 EDT 2022. Contains 354851 sequences. (Running on oeis4.)