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%I #10 Mar 10 2016 03:13:50
%S 7,-8,49,-49,121,-121,225,-225,361,-361,529,-529,729,-729,961,-961,
%T 1225,-1225,1521,-1521,1849,-1849,2209,-2209,2601,-2601,3025,-3025,
%U 3481,-3481,3969,-3969,4489,-4489,5041,-5041,5625,-5625,6241,-6241,6889,-6889,7569
%N First differences of number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 5", 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="/A270009/b270009.txt">Table of n, a(n) for n = 0..127</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_, Mar 09 2016: (Start)
%F a(n) = 4+5*(-1)^n+(4+8*(-1)^n)*n+4*(-1)^n*n^2 for n>1.
%F a(n) = 4*n^2+12*n+9 for n>1 and n even.
%F a(n) = -4*n^2-4*n-1 for n>1 and n odd.
%F a(n) = -a(n-1)+2*a(n-2)+2*a(n-3)-a(n-4)-a(n-5) for n>6.
%F G.f.: (7-x+27*x^2+2*x^3-3*x^4-x^5+x^6) / ((1-x)^2*(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=5; 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 Table[on[[i+1]]-on[[i]],{i,1,Length[on]-1}] (* Difference at each stage *)
%Y Cf. A270006.
%K sign,easy
%O 0,1
%A _Robert Price_, Mar 08 2016
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