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 A270009 First differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 5", based on the 5-celled von Neumann neighborhood. 2
 7, -8, 49, -49, 121, -121, 225, -225, 361, -361, 529, -529, 729, -729, 961, -961, 1225, -1225, 1521, -1521, 1849, -1849, 2209, -2209, 2601, -2601, 3025, -3025, 3481, -3481, 3969, -3969, 4489, -4489, 5041, -5041, 5625, -5625, 6241, -6241, 6889, -6889, 7569 (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 09 2016: (Start) a(n) = 4+5*(-1)^n+(4+8*(-1)^n)*n+4*(-1)^n*n^2 for n>1. a(n) = 4*n^2+12*n+9 for n>1 and n even. a(n) = -4*n^2-4*n-1 for n>1 and n odd. a(n) = -a(n-1)+2*a(n-2)+2*a(n-3)-a(n-4)-a(n-5) for n>6. G.f.: (7-x+27*x^2+2*x^3-3*x^4-x^5+x^6) / ((1-x)^2*(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=5; 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. A270006. Sequence in context: A303732 A119453 A047192 * A033044 A025630 A036566 Adjacent sequences:  A270006 A270007 A270008 * A270010 A270011 A270012 KEYWORD sign,easy AUTHOR Robert Price, Mar 08 2016 STATUS approved

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Last modified August 18 13:25 EDT 2019. Contains 326100 sequences. (Running on oeis4.)