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 A273480 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 737", based on the 5-celled von Neumann neighborhood. 1
 1, 5, 30, 79, 160, 281, 450, 675, 964, 1325, 1766, 2295, 2920, 3649, 4490, 5451, 6540, 7765, 9134, 10655, 12336, 14185, 16210, 18419, 20820, 23421, 26230, 29255, 32504, 35985, 39706, 43675, 47900, 52389, 57150, 62191, 67520, 73145, 79074, 85315, 91876, 98765 (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 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, May 23 2016: (Start) a(n) = (4*n^3+12*n^2+11*n-12)/3 for n>0. a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4. G.f.: (1+x+16*x^2-15*x^3+5*x^4) / (1-x)^4. (End) MATHEMATICA CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}]; code=737; 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[Total[Part[on, Range[1, i]]], {i, 1, Length[on]}] (* Sum at each stage *) CROSSREFS Cf. A016754. Sequence in context: A270811 A152745 A187275 * A164015 A128302 A258582 Adjacent sequences:  A273477 A273478 A273479 * A273481 A273482 A273483 KEYWORD nonn,easy AUTHOR Robert Price, May 23 2016 STATUS approved

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Last modified July 17 17:14 EDT 2019. Contains 325107 sequences. (Running on oeis4.)