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 A269906 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 1", based on the 5-celled von Neumann neighborhood. 4
 1, 4, 1, 44, 1, 116, 1, 220, 1, 356, 1, 524, 1, 724, 1, 956, 1, 1220, 1, 1516, 1, 1844, 1, 2204, 1, 2596, 1, 3020, 1, 3476, 1, 3964, 1, 4484, 1, 5036, 1, 5620, 1, 6236, 1, 6884, 1, 7564, 1, 8276, 1, 9020, 1, 9796, 1, 10604, 1, 11444, 1, 12316, 1, 13220, 1 (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 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, Mar 08 2016: (Start) a(n) = (-3+5*(-1)^n-4*(-1+(-1)^n)*n-4*(-1+(-1)^n)*n^2)/2. a(n) = 1 for n even. a(n) = 4*n^2+4*n-4 for n odd. a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>5. G.f.: (1+4*x-2*x^2+32*x^3+x^4-4*x^5) / ((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=1; 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}]; Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *) CROSSREFS Sequence in context: A369868 A302441 A373459 * A092667 A060627 A113101 Adjacent sequences: A269903 A269904 A269905 * A269907 A269908 A269909 KEYWORD nonn,easy,changed AUTHOR Robert Price, Mar 07 2016 STATUS approved

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Last modified August 7 14:45 EDT 2024. Contains 375012 sequences. (Running on oeis4.)