A137246 a(n) is the ratio of the sum of the squares of the bends (curvatures) of the n-th generation of an Apollonian packing to the sum of the squares of the bends of the initial four-circle configuration.

1, 17, 339, 6729, 133563, 2651073, 52620771, 1044462201, 20731381707, 411494247537, 8167690805619, 162119333369769, 3217883594978523, 63871313899461153, 1267772627204287491, 25163838602387366361, 499473454166134464747, 9913977567515527195857

Offset

1,2

Comments

These ratios are independent of the starting configuration. Similar ratios of third and higher moments are not so independent.

See A189226 for additional comments, references and links.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200 [a(188) corrected by Georg Fischer, May 24 2019]

J. C. Lagarias, C. L. Mallows, and Allan Wilks, Beyond the Descartes Circle Theorem, arXiv:math/0101066 [math.MG], 2001.

J. C. Lagarias, C. L. Mallows, and Allan Wilks, Beyond the Descartes Circle Theorem, Amer. Math Monthly, 109 (2002), 338-361.

C. L. Mallows, Growing Apollonian Packings, J. Integer Sequences, 12 (2009), article 09.2.1, page 3.

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

Formula

For n >= 4, a(n) = 20*a(n-1) - 3*a(n-2).

O.g.f.: x*(1-x)*(1-2*x)/(1-20*x+3*x^2). - R. J. Mathar, Mar 31 2008

a(n) = ((41+sqrt(97))*(10+sqrt(97))^(n-1) - (41-sqrt(97))*(10-sqrt(97))^(n-1))/(6*sqrt(97)) for n>1. - Bruno Berselli, Jul 04 2011

Example

Starting with the configuration with bends (-1,2,2,3) with sum(bends^2) = 18, the next generation contains four circles with bends 3,6,6,15. The sum of their squares is 306 = 18*a(2). The third generation has 12 circles with sum(bends^2) = 6102 = 18*a(3).

Mathematica

CoefficientList[Series[(2z^2-3z+1)/(3z^2-20z+1), {z, 0, 30}], z] (* and *) LinearRecurrence[{20, -3}, {1, 17, 339}, 30] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)

Prog

(PARI) Vec(x*(1-2*x)*(1-x)/(1-20*x+3*x^2)+O(x^30)) \\ Charles R Greathouse IV, Jul 03 2011
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!(x*(1-x)*(1-2*x)/(1-20*x+3*x^2)));  // Bruno Berselli, Jul 04 2011
(Sage) a=(x*(1-x)*(1-2*x)/(1-20*x+3*x^2)).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, May 24 2019
(GAP) a:=[1, 17, 339];; for n in [4..30] do a[n]:=20*a[n-1]-3*a[n-2]; od; a; # G. C. Greubel, May 24 2019

Crossrefs

Cf. A135849, A105970, A189226, A189227.

Sequence in context: A012112 A294435 A361096 * A171860 A324449 A191589

Adjacent sequences: A137243 A137244 A137245 * A137247 A137248 A137249

Keyword

easy,nonn

Author

Colin Mallows, Mar 09 2008

Status

approved