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 A270087 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 73", based on the 5-celled von Neumann neighborhood. 4
 1, 4, 5, 40, 0, 121, 0, 225, 0, 361, 0, 529, 0, 729, 0, 961, 0, 1225, 0, 1521, 0, 1849, 0, 2209, 0, 2601, 0, 3025, 0, 3481, 0, 3969, 0, 4489, 0, 5041, 0, 5625, 0, 6241, 0, 6889, 0, 7569, 0, 8281, 0, 9025, 0, 9801, 0, 10609, 0, 11449, 0, 12321, 0, 13225, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 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..128 Robert Price, Diagrams of the first 20 stages. 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) = -1/2*(-1+(-1)^n)*(1+2*n)^2 for n>3. a(n) = 0 for n>3 and even. a(n) = 4*n^2+4*n+1 for n>3 and odd. a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>9. G.f.: (1+4*x+2*x^2+28*x^3-12*x^4+13*x^5+14*x^6-22*x^7-5*x^8+9*x^9) / ((1-x)^3*(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=73; 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}]; Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *) CROSSREFS Sequence in context: A336024 A228798 A041557 * A265689 A270098 A271285 Adjacent sequences:  A270084 A270085 A270086 * A270088 A270089 A270090 KEYWORD nonn,easy AUTHOR Robert Price, Mar 10 2016 STATUS approved

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Last modified June 21 10:09 EDT 2021. Contains 345360 sequences. (Running on oeis4.)