A154636 a(n) is the ratio of the sum of the bends of the circles that are drawn in the n-th generation of Apollonian packing to the sum of the bends of the circles in the initial configuration of 3 circles.

1, 2, 18, 138, 1050, 7986, 60738, 461946, 3513354, 26720994, 203227890, 1545660138, 11755597434, 89407799058, 679995600162, 5171741404122, 39333944432490, 299156331247554, 2275248816682962, 17304521539721034, 131610425867719386, 1000969842322591986

Offset

0,2

Comments

For comments and more references and links, see A189226.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

C. L. Mallows, Growing Apollonian Packings, J. Integer Sequences, 12 (2009), article 09.2.1.

Index entries for linear recurrences with constant coefficients, signature (8,-3).

Formula

G.f.: (1 - x)*(1 - 5*x) / (1 - 8*x + 3*x^2).

From Colin Barker, Jul 15 2017: (Start)

a(n) = ((-(-7+sqrt(13))*(4+sqrt(13))^n - (4-sqrt(13))^n*(7+sqrt(13)))) / (3*sqrt(13)) for n>0.

a(n) = 8*a(n-1) - 3*a(n-2) for n>2.

(End)

Example

Starting from three circles with bends -1,2,2 summing to 3, the first derived generation consists of two circles, each with bend 3. So a(1) is (3+3)/3 = 2.

Mathematica

CoefficientList[Series[(5 z^2 - 6 z + 1)/(3 z^2 - 8 z + 1), {z, 0, 100}], z] (* and *) LinearRecurrence[{8, -3}, {1, 2, 18}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)

Prog

(PARI) Vec((1 - x)*(1 - 5*x) / (1 - 8*x + 3*x^2) + O(x^30)) \\ Colin Barker, Jul 15 2017

Crossrefs

Other sequences relating to the two-dimensional case are A135849, A137246, A154637. For the three-dim. case see A154638 - A154645. Five dimensions: A154635.

Cf. also A189226, A189227.

Sequence in context: A360132 A073512 A005544 * A362761 A216584 A193446

Adjacent sequences: A154633 A154634 A154635 * A154637 A154638 A154639

Keyword

easy,nonn

Author

Colin Mallows, Jan 13 2009

Extensions

More terms from N. J. A. Sloane, Nov 22 2009

Status

approved