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%I #26 Apr 03 2016 10:05:57
%S 1,4,5,36,9,96,17,188,21,312,25,468,29,656,33,876,37,1128,41,1412,45,
%T 1728,49,2076,53,2456,57,2868,61,3312,65,3788,69,4296,73,4836,77,5408,
%U 81,6012,85,6648,89,7316,93,8016,97,8748,101,9512,105,10308,109,11136
%N Number of active (ON,black) cells in n-th stage of growth of 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 A270569.
%D S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H Robert Price, <a href="/A269782/b269782.txt">Table of n, a(n) for n = 0..128</a>
%H Robert Price, <a href="/A269782/a269782.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 Conjectures from _Colin Barker_, Apr 03 2016: (Start)
%F a(n) = (11-(-1)^n+4*(-1)^n*n-4*(-1+(-1)^n)*n^2)/2 for n>4.
%F a(n) = 2*n+5 for n>4 and even.
%F a(n) = 4*n^2-2*n+6 for n>4 and odd.
%F a(n) = 3*a(n-2)-3*a(n-4)+a(n-6) for n>8.
%F G.f.: (1+4*x+2*x^2+24*x^3-3*x^4+4*x^6+4*x^7-8*x^8+4*x^10) / ((1-x)^3*(1+x)^3).
%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 Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%Y Cf. A270569.
%K nonn,easy
%O 0,2
%A _Robert Price_, Mar 10 2016
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