cp's OEIS Frontend

A277560 Binary representation of the x-axis, from the left edge to the origin, or from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 7", based on the 5-celled von Neumann neighborhood.

%I A277560 #45 Feb 16 2025 08:33:37
%S A277560 1,11,0,1111,0,111111,0,11111111,0,1111111111,0,111111111111,0,
%T A277560 11111111111111,0,1111111111111111,0,111111111111111111,0,
%U A277560 11111111111111111111,0,1111111111111111111111,0,111111111111111111111111,0,11111111111111111111111111,0
%N A277560 Binary representation of the x-axis, from the left edge to the origin, or from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 7", based on the 5-celled von Neumann neighborhood.
%C A277560 Initialized with a single black (ON) cell at stage zero.
%D A277560 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A277560 Robert Price, <a href="/A277560/b277560.txt">Table of n, a(n) for n = 0..126</a>
%H A277560 Robert Price, <a href="/A277560/a277560.tmp.txt">Diagrams of the first 20 stages</a>
%H A277560 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 A277560 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A277560 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A277560 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A277560 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A277560 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A277560 Conjectures from _Colin Barker_, Nov 06 2016: (Start)
%F A277560 G.f.: (1 + 11*x - 101*x^2 + 100*x^4)/((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).
%F A277560 a(n) = 101*a(n-2) - 100*a(n-4) for n>4.
%F A277560 a(n) = (-1)*(-1 + (-1)^n)*(-1 - 10^n)/18 for n>0. (End)
%t A277560 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A277560 code=7; stages=128;
%t A277560 rule=IntegerDigits[code,2,10];
%t A277560 g=2*stages+1; (* Maximum size of grid *)
%t A277560 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A277560 ca=a;
%t A277560 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A277560 PrependTo[ca,a];
%t A277560 (* Trim full grid to reflect growth by one cell at each stage *)
%t A277560 k=(Length[ca[[1]]]+1)/2;
%t A277560 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 A277560 Table[FromDigits[Part[ca[[i]][[i]],Range[1,i]],10], {i,1,stages-1}]
%Y A277560 Cf. A277936.
%K A277560 nonn,easy
%O A277560 0,2
%A A277560 _Robert Price_, Nov 05 2016