A302487 Total domination number of the n-Apollonian network.

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, 11438396227483, 34315188682444

Offset

1,1

Comments

For n > 1, also the feedback vertex set number of the (n-1)-Apollonian network. - Eric W. Weisstein, Feb 02 2026

Links

Table of n, a(n) for n=1..32.

Eric Weisstein's World of Mathematics, Apollonian Network.

Eric Weisstein's World of Mathematics, Feedback Vertex Set Number.

Eric Weisstein's World of Mathematics, Total Domination Number.

Index entries for linear recurrences with constant coefficients, signature (4,-3).

Formula

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

From Elmo R. Oliveira, Nov 22 2025: (Start)

G.f.: x*(2 - 6*x + x^2 - 2*x^3)/((3*x-1)*(x-1)).

E.g.f.: (exp(3*x) + 135*exp(x) - 136 - 30*x - 18*x^2)/54.

a(n) = 4*a(n-1) - 3*a(n-2) for n > 4. (End)

Mathematica

A302487[n_] := If[n <= 2, 2, (135 + 3^n)/54]; Array[A302487, 35] (* or *)
LinearRecurrence[{4, -3}, {2, 2, 3, 4}, 35] (* Paolo Xausa, Feb 26 2026 *)

Crossrefs

Cf. A291773, A347479.

Sequence in context: A023105 A281723 A011784 * A032252 A112708 A378753

Adjacent sequences: A302484 A302485 A302486 * A302488 A302489 A302490

Keyword

nonn,easy

Author

Eric W. Weisstein, Apr 08 2018

Extensions

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

Status

approved