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A137246
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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.
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6
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1, 17, 339, 6729, 133563, 2651073, 52620771, 1044462201, 20731381707, 411494247537, 8167690805619, 162119333369769, 3217883594978523, 63871313899461153, 1267772627204287491, 25163838602387366361, 499473454166134464747, 9913977567515527195857
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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.
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REFERENCES
| J. C. Lagarias, C. L. Mallows and A. R. Wilks, Beyond the Descartes Circle Theorem, Amer. Math. Monthly, 109 (2002), 338-361.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 1..200
J. C. Lagarias, C. L. Mallows, and A. R. Wilks, Beyond the Descartes Circle Theorem, Amer. Math Monthly, 109 (2002), 338-361.
Index entries for sequences related to linear recurrences with constant coefficients, signature (20,-3).
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FORMULA
| For n >= 4, a(n) = 20a(n-1) - 3a(n-2)
O.g.f.: x*(2*x-1)*(x-1)/(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
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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).
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MATHEMATICA
| CoefficientList[Series[(2 z^2 - 3 z + 1)/(3 z^2 - 20 z + 1), {z, 0, 100}], z] (* and *) LinearRecurrence[{20, -3}, {1, 17, 339}, 100] (* From Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)
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PROG
| (PARI) Vec(x*(1-2*x)*(1-x)/(1-20*x+3*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jul 03, 2011
(MAGMA) m:=19; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-x)*(1-2*x)/(1-20*x+3*x^2))); // Bruno Berselli, Jul 04 2011
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CROSSREFS
| Cf. A135849, A105970, A189226, A189227.
Sequence in context: A136270 A009046 A012112 * A171860 A191589 A194729
Adjacent sequences: A137243 A137244 A137245 * A137247 A137248 A137249
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KEYWORD
| easy,nonn
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AUTHOR
| Colin Mallows (colinm(AT)research.avayalabs.com), Mar 09 2008
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EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 31 2008
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