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
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Sep 01 2017
STATUS
approved