A154637 a(n) is the ratio of the sum of squares of the bends of the circles that are added in the n-th generation of Apollonian packing, to the sum of squares of the bends of the initial three circles.

1, 2, 66, 1314, 26082, 517698, 10275714, 203961186, 4048396578, 80356048002, 1594975770306, 31658447262114, 628384017931362, 12472705016840898, 247568948283023874, 4913960850609954786, 97536510167350024098, 1935988320795170617602, 38427156885401362279746, 762735172745641733742114

Offset

0,2

Comments

For more references and links, see A189226.

Links

Colin Barker, Table of n, a(n) for n = 0..750

Colin Mallows, Growing Apollonian packings, J. Integer Sequences v.12, article 09.2.1 (2009).

Index entries for linear recurrences with constant coefficients, signature (20,-3).

Formula

G.f.: (1-18*x+29*x^2) / (1-20*x+3*x^2).

From Colin Barker, Nov 16 2016: (Start)

a(n) = ((133-13*sqrt(97))*(10+sqrt(97))^n - (10-sqrt(97))^n*(133+13*sqrt(97))) / (3*sqrt(97)) for n>0.

a(n) = 20*a(n-1) - 3*a(n-2) for n>2.

(End)

Example

Starting with three circles with bends -1,2,2, the ssq is 9. The first derived generation has two circles, each with bend 3. So a(1) = (9+9)/9 = 2.

Mathematica

CoefficientList[Series[(29 z^2 - 18 z + 1)/(3 z^2 - 20 z + 1), {z, 0, 100}], z] (* and *) LinearRecurrence[{20, -3}, {1, 2, 66}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)

Prog

(PARI) Vec((1-18*x+29*x^2)/(1-20*x+3*x^2) + O(x^30)) \\ Colin Barker, Nov 16 2016

Crossrefs

For starting with four circles, see A137246. For sums of bends, see A135849 and A154636. For three dimensions, see A154638 - A154645.

Cf. also A189226, A189227.

Sequence in context: A098532 A159716 A157060 * A069865 A218433 A092884

Adjacent sequences: A154634 A154635 A154636 * A154638 A154639 A154640

Keyword

easy,nonn

Author

Colin Mallows, Jan 13 2009

Extensions

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

Status

approved