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

0,2

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

Table of n, a(n) for n=0..15.

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

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)

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 *)

Crossrefs

Cf. A269695.

Sequence in context: A166859 A167106 A167650 * A167896 A168682 A168730

Adjacent sequences: A269693 A269694 A269695 * A269697 A269698 A269699

Keyword

nonn,easy

Author

Robert Price, Mar 03 2016

Extensions

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

Status

approved