A273418 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 705", based on the 5-celled von Neumann neighborhood.
1, 4, 41, 217, 953, 3961, 16121, 65017, 261113, 1046521, 4190201, 16769017, 67092473, 268402681, 1073676281, 4294836217
Offset: 0
References
- S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
Links
- 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
Crossrefs
Cf. A273417.
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=705; 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}]; on=Map[Function[Apply[Plus,Flatten[#1]]],ca] (* Count ON cells at each stage *) Part[on,2^Range[0,Log[2,stages]]] (* Extract relevant terms *)
Formula
Conjecture: a(n) = A273313(n), n>1. - R. J. Mathar, May 27 2016
Conjecture: a(n) = 4*4^n - 4*2^n - 7, n>1. - Lars Blomberg, Jul 19 2016
Conjectures from Colin Barker, Dec 01 2016: (Start)
a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3) for n>4.
G.f.: (1 - 3*x + 27*x^2 - 22*x^3 - 24*x^4) / ((1 - x)*(1 - 2*x)*(1 - 4*x)).
(End)
Extensions
a(8)-a(15) from Lars Blomberg, Jul 19 2016
Comments