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A277926 Binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 5", based on the 5-celled von Neumann neighborhood.

%I A277926 #10 Feb 16 2025 08:33:37
%S A277926 1,10,0,1111,0,111111,0,11111111,0,1111111111,0,111111111111,0,
%T A277926 11111111111111,0,1111111111111111,0,111111111111111111,0,
%U A277926 11111111111111111111,0,1111111111111111111111,0,111111111111111111111111,0,11111111111111111111111111,0
%N A277926 Binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 5", based on the 5-celled von Neumann neighborhood.
%C A277926 Initialized with a single black (ON) cell at stage zero.
%D A277926 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A277926 Robert Price, <a href="/A277926/b277926.txt">Table of n, a(n) for n = 0..126</a>
%H A277926 Robert Price, <a href="/A277926/a277926.tmp.txt">Diagrams of first 20 stages</a>
%H A277926 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 A277926 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A277926 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A277926 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A277926 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A277926 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A277926 Robert Price, <a href="/A277926/a277926.tmp.txt">Diagrams of first 20 stages</a>
%F A277926 Conjectures from _Colin Barker_, Nov 04 2016: (Start)
%F A277926 a(n) = 0 for n>1 and even; a(n) = (10^(n+1)-1)/9 for n>1 and odd.
%F A277926 a(n) = 101*a(n-2)-100*a(n-4) for n>5.
%F A277926 G.f.: (1+10*x-101*x^2+101*x^3+100*x^4-100*x^5) / ((1-x)*(1+x)*(1-10*x)*(1+10*x)).
%F A277926 (End)
%t A277926 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A277926 code=5; stages=128;
%t A277926 rule=IntegerDigits[code,2,10];
%t A277926 g=2*stages+1; (* Maximum size of grid *)
%t A277926 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A277926 ca=a;
%t A277926 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A277926 PrependTo[ca,a];
%t A277926 (* Trim full grid to reflect growth by one cell at each stage *)
%t A277926 k=(Length[ca[[1]]]+1)/2;
%t A277926 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 A277926 Table[FromDigits[Part[ca[[i]][[i]],Range[1,i]],10], {i,1,stages-1}]
%Y A277926 Cf. A277927, A277928, A277929.
%K A277926 nonn,easy
%O A277926 0,2
%A A277926 _Robert Price_, Nov 04 2016