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A277955 Decimal 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 A277955 #45 Feb 16 2025 08:33:37
%S A277955 1,3,3,7,11,23,43,87,171,343,683,1367,2731,5463,10923,21847,43691,
%T A277955 87383,174763,349527,699051,1398103,2796203,5592407,11184811,22369623,
%U A277955 44739243,89478487,178956971,357913943,715827883,1431655767,2863311531,5726623063
%N A277955 Decimal 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 A277955 Initialized with a single black (ON) cell at stage zero.
%C A277955 Essentially the same as A267052. - _R. J. Mathar_, Nov 09 2016
%D A277955 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A277955 Robert Price, <a href="/A277955/b277955.txt">Table of n, a(n) for n = 0..126</a>
%H A277955 Robert Price, <a href="/A277955/a277955.tmp.txt">Diagrams of the first 20 stages</a>
%H A277955 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 A277955 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A277955 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A277955 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A277955 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A277955 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%F A277955 G.f.: (1 + x - 4*x^2)/(1 - 2*x - x^2 + 2*x^3). - _Robert G. Wilson v_, Nov 05 2016
%F A277955 From _Colin Barker_, Nov 06 2016: (Start)
%F A277955 a(n) = (3 - 2*(-1)^n + 2^(1+n))/3.
%F A277955 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. (End)
%F A277955 From _Paul Curtz_, May 08 2024: (Start)
%F A277955 a(2*n) = A007583(n). a(2*n+1) = A163834(n+1).
%F A277955 a(n) = A001045(n+1) + A010673(n).
%F A277955 a(n) = a(n-1) + 2*A078008(n-1). (End)
%t A277955 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A277955 code=14; stages=128;
%t A277955 rule=IntegerDigits[code,2,10];
%t A277955 g=2*stages+1; (* Maximum size of grid *)
%t A277955 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A277955 ca=a;
%t A277955 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A277955 PrependTo[ca,a];
%t A277955 (* Trim full grid to reflect growth by one cell at each stage *)
%t A277955 k=(Length[ca[[1]]]+1)/2;
%t A277955 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 A277955 Table[FromDigits[Part[ca[[i]][[i]],Range[i,2*i-1]],2], {i,1,stages-1}]
%t A277955 LinearRecurrence[{2, 1, -2}, {1, 3, 3}, 32] (* or *)
%t A277955 CoefficientList[ Series[(1 + x - 4x^2)/(1 - 2x - x^2 + 2x^3), {x, 0, 31}], x] (* _Robert G. Wilson v_, Nov 05 2016 *)
%o A277955 (Magma) I:=[1,3,3]; [n le 3 select I[n] else 2*Self(n-1)+Self(n-2)-2*Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Nov 06 2016
%Y A277955 Cf. A277952, A277953, A277954.
%Y A277955 Cf. A010673, A001045, A007583, A163834.
%Y A277955 Cf. A078008.
%K A277955 nonn,easy
%O A277955 0,2
%A A277955 _Robert Price_, Nov 05 2016