A269712 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 20", based on the 5-celled von Neumann neighborhood.
1, 4, 12, 28, 60, 124, 252, 508, 1020, 2044, 4092, 8188, 16380, 32764, 65532, 131068
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. A269711.
Programs
-
Mathematica
rule=20; stages=300; ca=CellularAutomaton[{rule,{2,{{0,2,0},{2,1,2},{0,2,0}}},{1,1}},{{{1}},0},stages]; (* Start with single black cell *) 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
Conjectures from Colin Barker, Mar 08 2016: (Start)
a(n) = 4*(2^n-1) =A028399(n+2) for n>0.
a(n) = 3*a(n-1)-2*a(n-2) for n>2.
G.f.: (1+x+2*x^2) / ((1-x)*(1-2*x)).
(End)
Extensions
a(9)-a(15) from Lars Blomberg, Apr 15 2016
Comments