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A277954 Decimal 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 14", based on the 5-celled von Neumann neighborhood.

%I A277954 #13 Feb 16 2025 08:33:37
%S A277954 1,3,6,14,26,58,106,234,426,938,1706,3754,6826,15018,27306,60074,
%T A277954 109226,240298,436906,961194,1747626,3844778,6990506,15379114,
%U A277954 27962026,61516458,111848106,246065834,447392426,984263338,1789569706,3937053354,7158278826
%N A277954 Decimal 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 14", based on the 5-celled von Neumann neighborhood.
%C A277954 Initialized with a single black (ON) cell at stage zero.
%D A277954 S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
%H A277954 Robert Price, <a href="/A277954/b277954.txt">Table of n, a(n) for n = 0..126</a>
%H A277954 Robert Price, <a href="/A277954/a277954.tmp.txt">Diagrams of the first 20 stages</a>
%H A277954 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 A277954 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>
%H A277954 S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>
%H A277954 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%H A277954 <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>
%H A277954 <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>
%H A277954 Robert Price, <a href="/A277954/a277954.tmp.txt">Diagrams of the first 20 stages</a>
%F A277954 Conjectures from _Colin Barker_, Nov 06 2016: (Start)
%F A277954 G.f.: (1+2*x-x^2) / ((1-x)*(1-2*x)*(1+2*x)).
%F A277954 a(n) = a(n-1)+4*a(n-2)-4*a(n-3) for n>2.
%F A277954 a(n) = (-8-(-2)^n+21*2^n)/12. (End)
%t A277954 CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}];
%t A277954 code=14; stages=128;
%t A277954 rule=IntegerDigits[code,2,10];
%t A277954 g=2*stages+1; (* Maximum size of grid *)
%t A277954 a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *)
%t A277954 ca=a;
%t A277954 ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}];
%t A277954 PrependTo[ca,a];
%t A277954 (* Trim full grid to reflect growth by one cell at each stage *)
%t A277954 k=(Length[ca[[1]]]+1)/2;
%t A277954 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 A277954 Table[FromDigits[Part[ca[[i]][[i]],Range[1,i]],2], {i,1,stages-1}]
%t A277954 LinearRecurrence[{1, 4, -4}, {1, 3, 6}, 31] (* or *)
%t A277954 CoefficientList[ Series[(1 + 2x - x^2)/(1 - x - 4x^2 + 4x^3), {x, 0, 31}], x] (* _Robert G. Wilson v_, Nov 05 2016 *)
%Y A277954 Cf. A277952, A277953, A277955.
%K A277954 nonn,easy
%O A277954 0,2
%A A277954 _Robert Price_, Nov 05 2016