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A289021
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Number of maximal independent vertex sets and minimal vertex covers in the n-Apollonian network.
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3
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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MATHEMATICA
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{1, 3} . # & /@ NestList[Function[{t, u}, {t^3 + u^3, t u^2}] @@ # &, {1, 1}, 6] (* Eric W. Weisstein, Sep 27 2017 *)
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PROG
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(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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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