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A273418 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 705", based on the 5-celled von Neumann neighborhood.

%I A273418 #16 Feb 16 2025 08:33:35
%S A273418 1,4,41,217,953,3961,16121,65017,261113,1046521,4190201,16769017,
%T A273418 67092473,268402681,1073676281,4294836217
%N A273418 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 705", based on the 5-celled von Neumann neighborhood.
%C A273418 Initialized with a single black (ON) cell at stage zero.
%C A273418 Conjecture: Rule 713 also generates this sequence. - _Lars Blomberg_, Jul 19 2016
%D A273418 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A273418 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 A273418 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A273418 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A273418 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A273418 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A273418 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A273418 Conjecture: a(n) = A273313(n), n>1. - _R. J. Mathar_, May 27 2016
%F A273418 Conjecture: a(n) = 4*4^n - 4*2^n - 7, n>1. - _Lars Blomberg_, Jul 19 2016
%F A273418 Conjectures from _Colin Barker_, Dec 01 2016: (Start)
%F A273418 a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>4.
%F A273418 G.f.: (1 - 3*x + 27*x^2 - 22*x^3 - 24*x^4) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).
%F A273418 (End)
%t A273418 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A273418 code=705; stages=128;
%t A273418 rule=IntegerDigits[code,2,10];
%t A273418 g=2*stages+1; (* Maximum size of grid *)
%t A273418 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A273418 ca=a;
%t A273418 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A273418 PrependTo[ca,a];
%t A273418 (* Trim full grid to reflect growth by one cell at each stage *)
%t A273418 k=(Length[ca[[1]]]+1)/2;
%t A273418 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 A273418 on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
%t A273418 Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)
%Y A273418 Cf. A273417.
%K A273418 nonn,more
%O A273418 0,2
%A A273418 _Robert Price_, May 22 2016
%E A273418 a(8)-a(15) from _Lars Blomberg_, Jul 19 2016