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.
1, 3, 6, 14, 26, 58, 106, 234, 426, 938, 1706, 3754, 6826, 15018, 27306, 60074, 109226, 240298, 436906, 961194, 1747626, 3844778, 6990506, 15379114, 27962026, 61516458, 111848106, 246065834, 447392426, 984263338, 1789569706, 3937053354, 7158278826
Offset: 0
References
- S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
Links
- Robert Price, Table of n, a(n) for n = 0..126
- Robert Price, Diagrams of the first 20 stages
- N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
- Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
- S. Wolfram, A New Kind of Science
- Index entries for sequences related to cellular automata
- Index to 2D 5-Neighbor Cellular Automata
- Index to Elementary Cellular Automata
- Robert Price, Diagrams of the first 20 stages
Programs
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Mathematica
CAStep[rule_,a_]:=Map[rule[[10-#]]&,ListConvolve[{{0,2,0},{2,1,2},{0,2,0}},a,2],{2}]; code=14; stages=128; rule=IntegerDigits[code,2,10]; g=2*stages+1; (* Maximum size of grid *) a=PadLeft[{{1}},{g,g},0,Floor[{g,g}/2]]; (* Initial ON cell on grid *) ca=a; ca=Table[ca=CAStep[rule,ca],{n,1,stages+1}]; PrependTo[ca,a]; (* Trim full grid to reflect growth by one cell at each stage *) k=(Length[ca[[1]]]+1)/2; ca=Table[Table[Part[ca[[n]][[j]],Range[k+1-n,k-1+n]],{j,k+1-n,k-1+n}],{n,1,k}]; Table[FromDigits[Part[ca[[i]][[i]],Range[1,i]],2], {i,1,stages-1}] LinearRecurrence[{1, 4, -4}, {1, 3, 6}, 31] (* or *) CoefficientList[ Series[(1 + 2x - x^2)/(1 - x - 4x^2 + 4x^3), {x, 0, 31}], x] (* Robert G. Wilson v, Nov 05 2016 *)
Formula
Conjectures from Colin Barker, Nov 06 2016: (Start)
G.f.: (1+2*x-x^2) / ((1-x)*(1-2*x)*(1+2*x)).
a(n) = a(n-1)+4*a(n-2)-4*a(n-3) for n>2.
a(n) = (-8-(-2)^n+21*2^n)/12. (End)
Comments