A347500 Number of dominating sets in the n-Apollonian network.
15, 109, 42953, 2960403451017, 1380531364206778111844580887042461529
Offset: 1
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..8
- Eric Weisstein's World of Mathematics, Apollonian Network.
- Eric Weisstein's World of Mathematics, Dominating Set.
Programs
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Mathematica
Map[ Sum[Binomial[3, k] #[[k + 1]] x^k, {k, 0, 3}] &, NestList[Function[{e0, e1, e2, e3}, {e0^3 + e1^3 x, e1^2 e0 + e2^2 e1 x, e2 e1^2 + e3 e2^2 x, e2^3 + e3^3 x}] @@ # &, {x, 1 + x, 1 + x, 1 + x}, 4] ] /. x -> 1 (* Eric W. Weisstein, Sep 03 2025 *) -
PARI
\\ here e0..e3 are for 0..3 outside vertices included in dominating set. step(S,x)={my([e0,e1,e2,e3]=S); [e0^3 + e1^3*x, e1^2*e0 + e2^2*e1*x, e2*e1^2 + e3*e2^2*x, e2^3 + e3^3*x]} a(n,x=1)={my(S=[x,1+x,1+x,1+x]); for(i=2, n, S=step(S,x)); sum(k=0, 3, binomial(3,k) * S[1+k] * x^k)} \\ Andrew Howroyd, May 29 2025
Extensions
a(4) onwards from Andrew Howroyd, May 29 2025
Comments