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 A273312 First differences of 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. 2
 3, 13, 23, 33, 39, 49, 55, 65, 71, 81, 87, 97, 103, 113, 119, 129, 135, 145, 151, 161, 167, 177, 183, 193, 199, 209, 215, 225, 231, 241, 247, 257, 263, 273, 279, 289, 295, 305, 311, 321, 327, 337, 343, 353, 359, 369, 375, 385, 391, 401, 407, 417, 423, 433 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Initialized with a single black (ON) cell at stage zero. Appears to be essentially the same as A273407. REFERENCES S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170. LINKS Robert Price, Table of n, a(n) for n = 0..127 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) = 8-(-1)^n+8*n for n>1. a(n) = 8*n+7 for n>1 and even. a(n) = 8*n+9 for n>1 and odd. a(n) = a(n-1)+a(n-2)-a(n-3) for n>2. G.f.: (3+10*x+7*x^2-4*x^4) / ((1-x)^2*(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)/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}]; on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *) Table[on[[i+1]]-on[[i]], {i, 1, Length[on]-1}] (* Difference at each stage *) CROSSREFS Cf. A273309, A273407. Sequence in context: A062667 A212525 A106099 * A017305 A018709 A121756 Adjacent sequences:  A273309 A273310 A273311 * A273313 A273314 A273315 KEYWORD nonn,easy AUTHOR Robert Price, May 19 2016 STATUS approved

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Last modified October 17 04:09 EDT 2019. Contains 328106 sequences. (Running on oeis4.)