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A273417 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 705", based on the 5-celled von Neumann neighborhood.

%I A273417 #15 Feb 16 2025 08:33:35
%S A273417 1,4,21,41,72,113,160,217,280,353,432,521,616,721,832,953,1080,1217,
%T A273417 1360,1513,1672,1841,2016,2201,2392,2593,2800,3017,3240,3473,3712,
%U A273417 3961,4216,4481,4752,5033,5320,5617,5920,6233,6552,6881,7216,7561,7912,8273,8640
%N A273417 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 705", based on the 5-celled von Neumann neighborhood.
%C A273417 Initialized with a single black (ON) cell at stage zero.
%D A273417 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A273417 Robert Price, <a href="/A273417/b273417.txt">Table of n, a(n) for n = 0..128</a>
%H A273417 Robert Price, <a href="/A273417/a273417.tmp.txt">Diagrams of the first 20 stages</a>
%H A273417 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 A273417 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A273417 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A273417 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A273417 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A273417 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A273417 Conjectures from _Colin Barker_, May 22 2016: (Start)
%F A273417 a(n) = (-15-(-1)^n+8*n+8*n^2)/2 for n>2.
%F A273417 a(n) = 4*(n^2+n-2) for n>2 and even.
%F A273417 a(n) = 4*n^2+4*n-7 for n>2 and odd.
%F A273417 a(n) = 2*a(n-1)-2*a(n-3)+a(n-4) for n>6.
%F A273417 G.f.: (1+2*x+13*x^2+x^3-3*x^4+7*x^5-5*x^6) / ((1-x)^3*(1+x)).
%F A273417 (End)
%t A273417 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A273417 code=705; stages=128;
%t A273417 rule=IntegerDigits[code,2,10];
%t A273417 g=2*stages+1; (* Maximum size of grid *)
%t A273417 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A273417 ca=a;
%t A273417 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A273417 PrependTo[ca,a];
%t A273417 (* Trim full grid to reflect growth by one cell at each stage *)
%t A273417 k=(Length[ca[[1]]]+1)/2;
%t A273417 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 A273417 Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%K A273417 nonn,easy
%O A273417 0,2
%A A273417 _Robert Price_, May 22 2016