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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. 4
1, 2, 66, 1314, 26082, 517698, 10275714, 203961186, 4048396578, 80356048002, 1594975770306, 31658447262114, 628384017931362, 12472705016840898, 247568948283023874, 4913960850609954786, 97536510167350024098, 1935988320795170617602, 38427156885401362279746, 762735172745641733742114 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 16 05:43 EDT 2019. Contains 328044 sequences. (Running on oeis4.)