

A289021


Number of maximal independent vertex sets and minimal vertex covers in the nApollonian network.


3




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^(n1). For n > 1, the size of the smallest such set, the independent domination number, is given by 3^(n2).
Also, for n > 1 the number of independent vertex sets and vertex covers in the (n1)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 A249113
Adjacent sequences: A289018 A289019 A289020 * A289022 A289023 A289024


KEYWORD

nonn


AUTHOR

Andrew Howroyd, Sep 01 2017


STATUS

approved



