%I #11 Jul 26 2024 21:16:42
%S 1,5,17,49,81,121,169,225,289,361,441,529,625,729,841,961,1089,1225,
%T 1369,1521,1681,1849,2025,2209,2401,2601,2809,3025,3249,3481,3721,
%U 3969,4225,4489,4761,5041,5329,5625,5929,6241,6561,6889,7225,7569,7921,8281,8649
%N Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 659", 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="/A273384/b273384.txt">Table of n, a(n) for n = 0..128</a>
%H Robert Price, <a href="/A273384/a273384.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_, May 21 2016: (Start)
%F a(n) = (1+2*n)^2 for n>2.
%F a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>5.
%F G.f.: (1+2*x+5*x^2+12*x^3-20*x^4+8*x^5) / (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=659; 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_, May 21 2016