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A291773
Domination number of the n-Apollonian network.
5
1, 1, 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
1,3
COMMENTS
Also, the connected domination number of the n-Apollonian network. - Andrew Howroyd, Jan 16 2018
LINKS
Eric Weisstein's World of Mathematics, Apollonian Network
Eric Weisstein's World of Mathematics, Connected Domination Number
Eric Weisstein's World of Mathematics, Domination Number
FORMULA
a(n) = (3^(n-3) + 5) / 2 for n > 2. - Andrew Howroyd, Sep 01 2017
From Colin Barker, Oct 03 2017: (Start)
G.f.: x*(1 - 3*x + 2*x^2 - 5*x^3) / ((1 - x)*(1 - 3*x)).
a(n) = 4*a(n-1) - 3*a(n-2) for n>4.
(End)
a(n) = A289521(n-3) for n > 3. - Andrew Howroyd, Jan 16 2018
MATHEMATICA
(* Start from Eric W. Weisstein, Jan 17 2018 *)
Join[{1, 1}, Table[(3^(n - 3) + 5)/2, {n, 3, 20}]]
Join[{1, 1}, Table[(3^n + 135)/54, {n, 3, 20}]]
Join[{1, 1}, (3^Range[3, 20] + 135)/54]
Join[{1, 1}, LinearRecurrence[{4, -3}, {3, 4}, 20]]
CoefficientList[Series[(1 - 3 x + 2 x^2 - 5 x^3)/(1 - 4 x + 3 x^2), {x, 0, 20}], x]
(* End *)
PROG
(PARI) \\ here d0..d3 are for 0..3 outside vertices included in dominating set.
D(d0, d1, d2, d3) = {[min(3*d0, 1+3*d1), min(d0+2*d1, 1+d1+2*d2), min(2*d1+d2, 1+2*d2+d3), min(3*d2, 1+3*d3)]}
a(n)={my(v=[1, 0, 0, 0]); for(i=2, n, v=D(v[1], v[2], v[3], v[4])); min(min(v[1], 1+v[2]), min(2+v[3], 3+v[4]))} \\ Andrew Howroyd, Sep 01 2017
(PARI) Vec(x*(1 - 3*x + 2*x^2 - 5*x^3) / ((1 - x)*(1 - 3*x)) + O(x^40)) \\ Colin Barker, Oct 03 2017
CROSSREFS
Cf. A298105.
Sequence in context: A291710 A100455 A363055 * A287752 A287493 A078825
KEYWORD
nonn,easy
AUTHOR
Eric W. Weisstein, Aug 31 2017
EXTENSIONS
a(7)-a(30) from Andrew Howroyd, Sep 01 2017
STATUS
approved