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 A271004 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 245", based on the 5-celled von Neumann neighborhood. 4
 1, 8, 4, 44, 13, 112, 25, 208, 37, 340, 49, 504, 61, 700, 73, 928, 85, 1188, 97, 1480, 109, 1804, 121, 2160, 133, 2548, 145, 2968, 157, 3420, 169, 3904, 181, 4420, 193, 4968, 205, 5548, 217, 6160, 229, 6804, 241, 7480, 253, 8188, 265, 8928, 277, 9700, 289 (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 28 2016: (Start) a(n) = (-13-9*(-1)^n+4*(2+(-1)^n)*n-4*(-1+(-1)^n)*n^2)/2 for n>5. a(n) = 6*n-11 for n>3 and even. a(n) = 4*n^2+2*n-2 for n>5 and odd. a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>9. G.f.: (1+8*x+x^2+20*x^3+4*x^4+4*x^5-3*x^6-4*x^7-3*x^8+8*x^9-4*x^11) / ((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=245; 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: A270677 A270901 A270934 * A271051 A046106 A112584 Adjacent sequences:  A271001 A271002 A271003 * A271005 A271006 A271007 KEYWORD nonn,easy AUTHOR Robert Price, Mar 28 2016 STATUS approved

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Last modified July 6 14:10 EDT 2022. Contains 355110 sequences. (Running on oeis4.)