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

1,1

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).

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

Prog

(PARI) a(n)=(3^n+5)/2 \\ Charles R Greathouse IV, Jul 07 2017
(PARI) Vec(x*(4 - 9*x) / ((1 - x)*(1 - 3*x)) + O(x^30)) \\ Colin Barker, Jul 07 2017

Crossrefs

Sequence in context: A050357 A296059 A132334 * A097661 A182561 A246915

Adjacent sequences: A289518 A289519 A289520 * A289522 A289523 A289524

Keyword

nonn,easy

Author

Andrew D. Walker, Jul 07 2017

Status

approved