A291773 -id:A291773 - cp's OEIS frontend

A298105 Number of connected dominating sets in the n-Apollonian network.

Original entry on oeis.org

15, 96, 30208, 1562294353920, 458842788759956138092569943104028672

Offset: 1

Eric W. Weisstein, Jan 12 2018

nonn

Term a(6) has 107 decimal digits and a(7) has 320 decimal digits. - Andrew Howroyd, Jan 16 2018

Andrew Howroyd, Table of n, a(n) for n = 1..7
Eric Weisstein's World of Mathematics, Apollonian Network
Eric Weisstein's World of Mathematics, Connected Dominating Set

Cf. A291773, A298123.

{1, 3, 3, 1} . # & /@ NestList[Function[{t0, t1, t2, t3}, {t1^3, t1 t2^2, t1^2 t2 + t2^2 t3, t2^3 + t3^3}] @@ # &, {1, 2, 2, 2}, 4] (* Eric W. Weisstein, Jan 17 2018 *)

\\ here t0..t3 are for 0..3 outside vertices included in set.
D(t0,t1,t2,t3)={[t1^3, t1*t2^2, t1^2*t2 + t2^2*t3, t2^3 + t3^3]}
a(n)={my(v=[1,2,2,2]); for(i=2, n, v=D(v[1], v[2], v[3], v[4])); v[1]+3*v[2]+3*v[3]+v[4]}

Terms a(4) and beyond from Andrew Howroyd, Jan 16 2018

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

Original entry on oeis.org

15, 109, 42953, 2960403451017, 1380531364206778111844580887042461529

Offset: 1

Eric W. Weisstein, Sep 04 2021

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

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.

Cf. A291773, A298105, A347479, A347511.

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 *)

\\ 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

a(4) onwards from Andrew Howroyd, May 29 2025

A289021 Number of maximal independent vertex sets and minimal vertex covers in the n-Apollonian network.

Original entry on oeis.org

4, 5, 15, 845, 403227665, 64175114443109790962237345, 264288160993294964501375691029638701718807009656135518176301450923295365341665

Offset: 1

Andrew Howroyd, Sep 01 2017

nonn

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.

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

Cf. A291773.

{1, 3} . # & /@ NestList[Function[{t, u}, {t^3 + u^3, t u^2}] @@ # &, {1, 1}, 6] (* Eric W. Weisstein, Sep 27 2017 *)

\\ 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);

A302487 Total domination number of the n-Apollonian network.

Original entry on oeis.org

2, 2, 3, 4, 7, 16, 43, 124, 367, 1096, 3283, 9844, 29527, 88576, 265723, 797164, 2391487, 7174456, 21523363, 64570084, 193710247, 581130736, 1743392203, 5230176604, 15690529807, 47071589416, 141214768243, 423644304724, 1270932914167, 3812798742496

Offset: 1

Eric W. Weisstein, Apr 08 2018

Eric Weisstein's World of Mathematics, Apollonian Network.
Eric Weisstein's World of Mathematics, Total Domination Number.
Index entries for linear recurrences with constant coefficients, signature (4,-3).

Cf. A291773, A347479.

a(n) = (3^(n-3) + 5) / 2 = A291773(n) for n >= 3. - Andrew Howroyd, May 29 2025

a(8)-a(9) corrected and a(10) onwards from Andrew Howroyd, May 29 2025

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A298105 Number of connected dominating sets in the n-Apollonian network.

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

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A289021 Number of maximal independent vertex sets and minimal vertex covers in the n-Apollonian network.

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A302487 Total domination number of the n-Apollonian network.

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