A289021 Number of maximal independent vertex sets and minimal vertex covers in the n-Apollonian network.

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

Andrew Howroyd, Table of n, a(n) for n = 1..9

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

Adjacent sequences: A289018 A289019 A289020 * A289022 A289023 A289024

Keyword

nonn

Author

Andrew Howroyd, Sep 01 2017

Status

approved