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
KEYWORD
easy,nonn
AUTHOR
Colin Mallows, Mar 09 2008
STATUS
approved