A289521 Number of vertices in a planar Apollonian graph at iteration n.
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
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- Eric Weisstein's World of Mathematics, Apollonian Network
- Index entries for linear recurrences with constant coefficients, signature (4,-3).
Programs
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PARI
a(n)=(3^n+5)/2 \\ Charles R Greathouse IV, Jul 07 2017
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PARI
Vec(x*(4 - 9*x) / ((1 - x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Jul 07 2017
Formula
From Colin Barker, Jul 07 2017: (Start)
G.f.: x*(4 - 9*x) / ((1 - x)*(1 - 3*x)).
a(n) = (5 + 3^n) / 2.
a(n) = 4*a(n-1) - 3*a(n-2) for n>2.
(End)
a(n) = a(n-1) + 3^(n-1) for n>1. - Andrew D. Walker, Jul 07 2017