%I #24 Feb 16 2025 08:33:51
%S 1,1,3,4,7,16,43,124,367,1096,3283,9844,29527,88576,265723,797164,
%T 2391487,7174456,21523363,64570084,193710247,581130736,1743392203,
%U 5230176604,15690529807,47071589416,141214768243,423644304724,1270932914167,3812798742496
%N Domination number of the n-Apollonian network.
%C Also, the connected domination number of the n-Apollonian network. - _Andrew Howroyd_, Jan 16 2018
%H Colin Barker, <a href="/A291773/b291773.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ApollonianNetwork.html">Apollonian Network</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ConnectedDominationNumber.html">Connected Domination Number</a>
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DominationNumber.html">Domination Number</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).
%F a(n) = (3^(n-3) + 5) / 2 for n > 2. - _Andrew Howroyd_, Sep 01 2017
%F From _Colin Barker_, Oct 03 2017: (Start)
%F G.f.: x*(1 - 3*x + 2*x^2 - 5*x^3) / ((1 - x)*(1 - 3*x)).
%F a(n) = 4*a(n-1) - 3*a(n-2) for n>4.
%F (End)
%F a(n) = A289521(n-3) for n > 3. - _Andrew Howroyd_, Jan 16 2018
%t (* Start from _Eric W. Weisstein_, Jan 17 2018 *)
%t Join[{1, 1}, Table[(3^(n - 3) + 5)/2, {n, 3, 20}]]
%t Join[{1, 1}, Table[(3^n + 135)/54, {n, 3, 20}]]
%t Join[{1, 1}, (3^Range[3, 20] + 135)/54]
%t Join[{1, 1}, LinearRecurrence[{4, -3}, {3, 4}, 20]]
%t CoefficientList[Series[(1 - 3 x + 2 x^2 - 5 x^3)/(1 - 4 x + 3 x^2), {x, 0, 20}], x]
%t (* End *)
%o (PARI) \\ here d0..d3 are for 0..3 outside vertices included in dominating set.
%o 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)]}
%o 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
%o (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
%Y Cf. A298105.
%K nonn,easy,changed
%O 1,3
%A _Eric W. Weisstein_, Aug 31 2017
%E a(7)-a(30) from _Andrew Howroyd_, Sep 01 2017