A347500 Number of dominating sets in the n-Apollonian network.

15, 109, 42953, 2960403451017, 1380531364206778111844580887042461529

Offset: 1

Author

Eric W. Weisstein, Sep 04 2021

Keywords

• nonn
• changed

Comments

Term a(6) has 108 decimal digits and a(7) has 323 decimal digits. - Andrew Howroyd, May 29 2025

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.

Crossrefs

Cf. A291773, A298105, A347479, A347511.

Programs

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