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 A273309 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 641", based on the 5-celled von Neumann neighborhood. 4
 1, 4, 17, 40, 73, 112, 161, 216, 281, 352, 433, 520, 617, 720, 833, 952, 1081, 1216, 1361, 1512, 1673, 1840, 2017, 2200, 2393, 2592, 2801, 3016, 3241, 3472, 3713, 3960, 4217, 4480, 4753, 5032, 5321, 5616, 5921, 6232, 6553, 6880, 7217, 7560, 7913, 8272, 8641 (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, May 19 2016: (Start) a(n) = (-15+(-1)^n+8*n+8*n^2)/2 for n>1. a(n) = 4*n^2+4*n-7 for n>1 and even. a(n) = 4*n^2+4*n-8 for n>1 and odd. a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>3. G.f.: (1+2*x+9*x^2+8*x^3-4*x^5) / ((1-x)^3*(1+x)). (End) MATHEMATICA CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code=641; 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: A178947 A041859 A022266 * A145995 A275815 A018973 Adjacent sequences:  A273306 A273307 A273308 * A273310 A273311 A273312 KEYWORD nonn,easy AUTHOR Robert Price, May 19 2016 STATUS approved

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Last modified June 1 09:53 EDT 2020. Contains 334762 sequences. (Running on oeis4.)