A271060 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 261", based on the 5-celled von Neumann neighborhood.
1, 8, 1, 48, 1, 120, 1, 224, 1, 360, 1, 528, 1, 728, 1, 960, 1, 1224, 1, 1520, 1, 1848, 1, 2208, 1, 2600, 1, 3024, 1, 3480, 1, 3968, 1, 4488, 1, 5040, 1, 5624, 1, 6240, 1, 6888, 1, 7568, 1, 8280, 1, 9024, 1, 9800, 1, 10608, 1, 11448, 1, 12320, 1, 13224, 1
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..128
- 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
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=261; 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}]; Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *)
Formula
Conjectures from Chai Wah Wu, Dec 29 2016: (Start)
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6) for n > 5.
G.f.: (-x^4 - 24*x^3 + 2*x^2 - 8*x - 1)/((x - 1)^3*(x + 1)^3). (End)
Comments