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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
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.
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
KEYWORD
nonn,easy
AUTHOR
Robert Price, Mar 03 2016
EXTENSIONS
a(9)-a(15) from Lars Blomberg, Apr 12 2016
STATUS
approved