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%I A273833 #12 Jul 26 2024 21:16:43
%S A273833 1,5,26,71,147,264,428,649,933,1290,1726,2251,2871,3596,4432,5389,
%T A273833 6473,7694,9058,10575,12251,14096,16116,18321,20717,23314,26118,29139,
%U A273833 32383,35860,39576,43541,47761,52246,57002,62039,67363,72984,78908,85145,91701,98586
%N A273833 Partial sums of the number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 961", based on the 5-celled von Neumann neighborhood.
%C A273833 Initialized with a single black (ON) cell at stage zero.
%D A273833 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A273833 Robert Price, <a href="/A273833/b273833.txt">Table of n, a(n) for n = 0..128</a>
%H A273833 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 A273833 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A273833 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A273833 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A273833 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A273833 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A273833 Conjectures from _Colin Barker_, Jun 01 2016: (Start)
%F A273833 a(n) = (16*n^3+48*n^2-10*n-3*((-1)^n-5))/12 for n>1.
%F A273833 a(n) = (8*n^3+24*n^2-5*n+6)/6 for n>1 and even.
%F A273833 a(n) = (8*n^3+24*n^2-5*n+9)/6 for n>1 and odd.
%F A273833 a(n) = 3*a(n-1)-2*a(n-2)-2*a(n-3)+3*a(n-4)-a(n-5) for n>6.
%F A273833 G.f.: (1+2*x+13*x^2+5*x^3-7*x^4+3*x^5-x^6) / ((1-x)^4*(1+x)).
%F A273833 (End)
%t A273833 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A273833 code=961; stages=128;
%t A273833 rule=IntegerDigits[code,2,10];
%t A273833 g=2*stages+1; (* Maximum size of grid *)
%t A273833 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A273833 ca=a;
%t A273833 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A273833 PrependTo[ca,a];
%t A273833 (* Trim full grid to reflect growth by one cell at each stage *)
%t A273833 k=(Length[ca[[1]]]+1)/2;
%t A273833 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 A273833 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A273833 Table[Total[Part[on,Range[1,i]]],{i,1,Length[on]}] (* Sum at each stage *)
%Y A273833 Cf. A273831.
%K A273833 nonn,easy
%O A273833 0,2
%A A273833 _Robert Price_, May 31 2016

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