|
%I #12 Mar 11 2016 10:13:41
%S 1,5,10,50,50,171,171,396,396,757,757,1286,1286,2015,2015,2976,2976,
%T 4201,4201,5722,5722,7571,7571,9780,9780,12381,12381,15406,15406,
%U 18887,18887,22856,22856,27345,27345,32386,32386,38011,38011,44252,44252,51141,51141
%N Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 73", 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="/A270089/b270089.txt">Table of n, a(n) for n = 0..128</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 11 2016: (Start)
%F a(n) = (-87-9*(-1)^n+(22-24*(-1)^n)*n-12*(-2+(-1)^n)*n^2+8*n^3)/12 for n>2.
%F a(n) = (4*n^3+6*n^2-n-48)/6 for n>2 and even.
%F a(n) = (4*n^3+18*n^2+23*n-39)/6 for n>2 and odd.
%F a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7) for n>9.
%F G.f.: (1+4*x+2*x^2+28*x^3-12*x^4+13*x^5+14*x^6-22*x^7-5*x^8+9*x^9) / ((1-x)^4*(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=73; 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[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)
%Y Cf. A270087.
%K nonn,easy
%O 0,2
%A _Robert Price_, Mar 10 2016
|
