%I #24 Feb 16 2025 08:33:48
%S 4,7,16,43,124,367,1096,3283,9844,29527,88576,265723,797164,2391487,
%T 7174456,21523363,64570084,193710247,581130736,1743392203,5230176604,
%U 15690529807,47071589416,141214768243,423644304724,1270932914167,3812798742496,11438396227483,34315188682444
%N Number of vertices in a planar Apollonian graph at iteration n.
%H Colin Barker, <a href="/A289521/b289521.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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3).
%F From _Colin Barker_, Jul 07 2017: (Start)
%F G.f.: x*(4 - 9*x) / ((1 - x)*(1 - 3*x)).
%F a(n) = (5 + 3^n) / 2.
%F a(n) = 4*a(n-1) - 3*a(n-2) for n>2.
%F (End)
%F a(n) = a(n-1) + 3^(n-1) for n>1. - _Andrew D. Walker_, Jul 07 2017
%o (PARI) a(n)=(3^n+5)/2 \\ _Charles R Greathouse IV_, Jul 07 2017
%o (PARI) Vec(x*(4 - 9*x) / ((1 - x)*(1 - 3*x)) + O(x^30)) \\ _Colin Barker_, Jul 07 2017
%K nonn,easy,changed
%O 1,1
%A _Andrew D. Walker_, Jul 07 2017