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%I #16 May 09 2016 23:58:40
%S 1,4,36,188,876,3788,15756,64268,259596,1043468,4184076,16756748,
%T 67067916,268353548,1073577996,4294639628
%N Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 65", based on the 5-celled von Neumann neighborhood.
%C Initialized with a single black (ON) cell at stage zero.
%C Similar to A271254.
%C _Lars Blomberg_ conjectured that Rule 321 also produces this sequence. It would be nice to have a proof.
%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%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 Conjecture: a(n) = 4*4^n - 10*2^n + 12, n>1. - _Lars Blomberg_, Apr 21 2016
%F Conjectures from _Colin Barker_, Apr 21 2016: (Start)
%F a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>4.
%F G.f.: (1-3*x+22*x^2-16*x^3+32*x^4) / ((1-x)*(1-2*x)*(1-4*x)).
%F (End)
%t CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t code=65; 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 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)
%Y Cf. A269782, A271254.
%K nonn,more
%O 0,2
%A _Robert Price_, Mar 10 2016
%E a(8)-a(15) from _Lars Blomberg_, Apr 21 2016
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