cp's OEIS Frontend

A277953 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 14", based on the 5-celled von Neumann neighborhood.

%I A277953 #15 Feb 16 2025 08:33:37
%S A277953 1,11,11,111,1011,10111,101011,1010111,10101011,101010111,1010101011,
%T A277953 10101010111,101010101011,1010101010111,10101010101011,
%U A277953 101010101010111,1010101010101011,10101010101010111,101010101010101011,1010101010101010111,10101010101010101011
%N A277953 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 14", based on the 5-celled von Neumann neighborhood.
%C A277953 Initialized with a single black (ON) cell at stage zero.
%C A277953 Essentially the same as A267051. - _R. J. Mathar_, Nov 09 2016
%D A277953 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A277953 Robert Price, <a href="/A277953/b277953.txt">Table of n, a(n) for n = 0..126</a>
%H A277953 Robert Price, <a href="/A277953/a277953.tmp.txt">Diagrams of the first 20 stages</a>
%H A277953 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 A277953 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A277953 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A277953 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A277953 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A277953 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A277953 Conjectures from _Colin Barker_, Nov 06 2016: (Start)
%F A277953 G.f.: (1+x-100*x^2) / ((1-x)*(1+x)*(1-10*x)).
%F A277953 a(n) = 10*a(n-1)+a(n-2)-10*a(n-3) for n>2.
%F A277953 a(n) = (539-450*(-1)^n+10^(1+n))/99. (End)
%t A277953 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A277953 code=14; stages=128;
%t A277953 rule=IntegerDigits[code,2,10];
%t A277953 g=2*stages+1; (* Maximum size of grid *)
%t A277953 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A277953 ca=a;
%t A277953 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A277953 PrependTo[ca,a];
%t A277953 (* Trim full grid to reflect growth by one cell at each stage *)
%t A277953 k=(Length[ca[[1]]]+1)/2;
%t A277953 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 A277953 Table[FromDigits[Part[ca[[i]][[i]],Range[i,2*i-1]],10], {i,1,stages-1}]
%Y A277953 Cf. A277952, A277954, A277955.
%K A277953 nonn,easy
%O A277953 0,2
%A A277953 _Robert Price_, Nov 05 2016