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 A271053 First differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 253", based on the 5-celled von Neumann neighborhood. 1
 7, -4, 40, -31, 103, -103, 207, -207, 343, -343, 511, -511, 711, -711, 943, -943, 1207, -1207, 1503, -1503, 1831, -1831, 2191, -2191, 2583, -2583, 3007, -3007, 3463, -3463, 3951, -3951, 4471, -4471, 5023, -5023, 5607, -5607, 6223, -6223, 6871, -6871, 7551 (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, Nov 22 2017: (Start) G.f.: (7 + 3*x + 22*x^2 + 3*x^3 + 7*x^4 - 15*x^5 - 4*x^6 + 9*x^7) / ((1 - x)^2*(1 + x)^3). a(n) = 4*n^2 + 12*n - 9 for n>2 and even. a(n) = -4*n^2 - 4*n + 17 for n>2 and odd. a(n) = -a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4) - a(n-5) for n>7. (End) MATHEMATICA CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code=253; 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. A271051. Sequence in context: A270983 A278047 A271007 * A270629 A270680 A270904 Adjacent sequences:  A271050 A271051 A271052 * A271054 A271055 A271056 KEYWORD sign,easy AUTHOR Robert Price, Mar 29 2016 STATUS approved

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Last modified September 22 11:06 EDT 2020. Contains 337289 sequences. (Running on oeis4.)