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%I #19 Jul 26 2024 21:16:38
%S 1,5,13,32,37,89,80,145,113,292,169,357,244,581,293,688,397,969,472,
%T 1105,585,1476,673,1621,828,2069,909,2256,1093,2761,1200,2977,1393,
%U 3572,1513,3797,1748,4469,1861,4736,2125,5465,2264,5761,2537,6580,2689,6885
%N Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 427", based on the 5-celled von Neumann neighborhood.
%C Initialized with a single black (ON) cell at stage zero.
%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H Robert Price, <a href="/A271902/b271902.txt">Table of n, a(n) for n = 0..128</a>
%H Robert Price, <a href="/A271902/a271902.tmp.txt">Diagrams of the first 20 stages</a>
%H N. J. A. Sloane, <a href="http://arxiv.org/abs/1503.01168">On the Number of ON Cells in Cellular Automata</a>, arXiv:1503.01168 [math.CO], 2015
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F Empirical g.f.: (1 +6*x +18*x^2 +44*x^3 +61*x^4 +96*x^5 +88*x^6 +68*x^7 +7*x^8 +26*x^9 +6*x^10 -5*x^12) / ((1 -x)^3*(1 +x)^3*(1 +x^2)^2*(1 +x +x^2)). - _Colin Barker_, Apr 21 2016
%t CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t code=427; stages=128;
%t rule=IntegerDigits[code,2,10];
%t g=2*stages+1; (* Maximum size of grid *)
%t a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t ca=a;
%t ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t PrependTo[ca,a];
%t (* Trim full grid to reflect growth by one cell at each stage *)
%t k=(Length[ca[[1]]]+1)/2;
%t ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}];
%t Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%K nonn,easy
%O 0,2
%A _Robert Price_, Apr 20 2016
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