A269708 Number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 14", based on the 5-celled von Neumann neighborhood.

1, 5, 20, 76, 292, 1132, 4420, 17356, 68452, 270892, 1074820, 4273036, 17013412, 67817452, 270561220, 1080119116

Offset: 0

Author

Robert Price, Mar 04 2016

Keywords

• nonn
• more

Comments

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

Rules 46, 142 and 174 also generate this sequence.

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. A269707.

Programs

Mathematica

rule=14; 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*3^(n-2)+4^n for n>1.

a(n) = 7*a(n-1)-12*a(n-2) for n>3.

G.f.: (1-2*x-3*x^2-4*x^3) / ((1-3*x)*(1-4*x)).

(End)

Extensions

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