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 A270086 First differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 65", based on the 5-celled von Neumann neighborhood. 2
 3, 1, 31, -27, 87, -79, 171, -167, 291, -287, 443, -439, 627, -623, 843, -839, 1091, -1087, 1371, -1367, 1683, -1679, 2027, -2023, 2403, -2399, 2811, -2807, 3251, -3247, 3723, -3719, 4227, -4223, 4763, -4759, 5331, -5327, 5931, -5927, 6563, -6559, 7227 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Initialized with a single black (ON) cell at stage zero. Similar to A271256. 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 11 2016: (Start) a(n) = 2+(-1)^n+4*n+4*(-1)^n*n^2 for n>4. a(n) = 4*n^2+4*n+3 for n>4 and even. a(n) = -4*n^2+4*n+1 for n>4 and odd. a(n) = -a(n-1)+2*a(n-2)+2*a(n-3)-a(n-4)-a(n-5) for n>7. G.f.: (3+4*x+26*x^2-4*x^3-x^4+4*x^5+4*x^6-8*x^7+4*x^9) / ((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=65; 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. A269782, A271256. Sequence in context: A326797 A227953 A046979 * A141411 A016481 A303818 Adjacent sequences:  A270083 A270084 A270085 * A270087 A270088 A270089 KEYWORD sign,easy AUTHOR Robert Price, Mar 10 2016 STATUS approved

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Last modified April 17 09:29 EDT 2021. Contains 343064 sequences. (Running on oeis4.)