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 A270082 First differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 62", based on the 5-celled von Neumann neighborhood. 1
 4, 7, 8, 12, 12, 24, 4, 28, 12, 44, 4, 44, 12, 60, 4, 60, 12, 76, 4, 76, 12, 92, 4, 92, 12, 108, 4, 108, 12, 124, 4, 124, 12, 140, 4, 140, 12, 156, 4, 156, 12, 172, 4, 172, 12, 188, 4, 188, 12, 204, 4, 204, 12, 220, 4, 220, 12, 236, 4, 236, 12, 252, 4, 252 (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 10 2016: (Start) a(n) = a(n-2)+a(n-4)-a(n-6) for n>9. G.f.: (4+7*x+4*x^2+5*x^3+5*x^5-12*x^6-x^7+4*x^8+4*x^9-4*x^11) / ((1-x)^2*(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=62; 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. A270079. Sequence in context: A061932 A270336 A270941 * A189223 A060833 A162967 Adjacent sequences:  A270079 A270080 A270081 * A270083 A270084 A270085 KEYWORD nonn,easy AUTHOR Robert Price, Mar 10 2016 STATUS approved

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Last modified April 20 12:58 EDT 2021. Contains 343135 sequences. (Running on oeis4.)