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 A270684 First differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 190", based on the 5-celled von Neumann neighborhood. 1
 4, 7, 12, 8, 20, 8, 28, 8, 36, 8, 44, 8, 52, 8, 60, 8, 68, 8, 76, 8, 84, 8, 92, 8, 100, 8, 108, 8, 116, 8, 124, 8, 132, 8, 140, 8, 148, 8, 156, 8, 164, 8, 172, 8, 180, 8, 188, 8, 196, 8, 204, 8, 212, 8, 220, 8, 228, 8, 236, 8, 244, 8, 252, 8, 260, 8, 268, 8 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 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..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, Mar 21 2016: (Start) a(n) = 2*(3-(-1)^n+n+(-1)^n*n) for n>1. a(n) = 4*n+4 for n>1 and even. a(n) = 8 for n>1 and odd. a(n) = 2*a(n-2)-a(n-4) for n>3. G.f.: (4+7*x+4*x^2-6*x^3-x^5) / ((1-x)^2*(1+x)^2). (End) MATHEMATICA CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code=190; 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}]; 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. A270681. Sequence in context: A061956 A178939 A072732 * A083487 A253129 A249918 Adjacent sequences:  A270681 A270682 A270683 * A270685 A270686 A270687 KEYWORD nonn,easy AUTHOR Robert Price, Mar 21 2016 STATUS approved

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