A269696 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 6", based on the 5-celled von Neumann neighborhood.

1, 5, 20, 80, 320, 1280, 5120, 20480, 81920, 327680, 1310720, 5242880, 20971520, 83886080, 335544320, 1342177280

Offset: 0

Author

Robert Price, Mar 03 2016

Keywords

• nonn
• easy

Comments

Initialized with a single black (ON) cell at stage zero.

Rules 38, 70, 102, 134, 166, 198 and 230 also generate this sequence.

Apparently a duplicate of A003947. - R. J. Mathar, Mar 09 2016

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. A269695.

Programs

Mathematica

rule=6; 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) = 5*4^(n-1) for n>0.

a(n) = 4*a(n-1) for n>1.

G.f.: (1+x) / (1-4*x).

(End)

Extensions

a(9)-a(15) from Lars Blomberg, Apr 12 2016