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A289021
Number of maximal independent vertex sets and minimal vertex covers in the n-Apollonian network.
3
4, 5, 15, 845, 403227665, 64175114443109790962237345, 264288160993294964501375691029638701718807009656135518176301450923295365341665
OFFSET
1,1
COMMENTS
Term a(8) has 233 decimal digits.
The size of the largest maximal independent vertex set, the independence number, is given by 3^(n-1). For n > 1, the size of the smallest such set, the independent domination number, is given by 3^(n-2).
Also, for n > 1 the number of independent vertex sets and vertex covers in the (n-1)-Apollonian network.
LINKS
Eric Weisstein's World of Mathematics, Apollonian Network
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set
Eric Weisstein's World of Mathematics, Minimal Vertex Cover
Eric Weisstein's World of Mathematics, Vertex Cover
MATHEMATICA
{1, 3} . # & /@ NestList[Function[{t, u}, {t^3 + u^3, t u^2}] @@ # &, {1, 1}, 6] (* Eric W. Weisstein, Sep 27 2017 *)
PROG
(PARI) \\ here t0..t1 are for 0..1 outside vertices included in set
T(t0, t1, x) = {[t0^3+t1^3*x, t0*t1^2]}
p(n, x)={my(v=[x, 1]); for(i=2, n, v=T(v[1], v[2], x)); v[1]+3*v[2]*x}
a(n)=p(n, 1);
CROSSREFS
Cf. A291773.
Sequence in context: A051721 A050226 A119562 * A323627 A289742 A340850
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Sep 01 2017
STATUS
approved