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A135849
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a(n) is the ratio of the sum of the bends (curvatures) of the circles in the n-th generation of an Apollonian packing to the sum of the bends in the initial four-circle configuration.
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6
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1, 5, 39, 297, 2259, 17181, 130671, 993825, 7558587, 57487221, 437222007, 3325314393, 25290849123, 192350849805, 1462934251071, 11126421459153, 84622568920011, 643601286982629, 4894942589100999, 37228736851860105, 283145067047577843, 2153474325825042429
(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.
For more comments, references and links, see A189226.
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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
| Index entries for sequences related to linear recurrences with constant coefficients, signature (8,-3).
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FORMULA
| For n >= 4, a(n)=8a(n-1)-3a(n-2)
For n>2, [a(n+2), a(n+3)] = the 2 X 2 matrix [0,1; -3,8] * [5,39]. Example: [0,1; -3,8]^3 * [5,39] = [a(5), a(6)] = [2259, 17181]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 09 2008
a(n) = floor(C * A138264(n)), where C = 1.057097576... = (1/2)*((1/9) + sqrt((1/81) + 4)). Example: a(7) = 130671 = floor(C * A138264(7)) = floor(C * 123613). A135849(n)/A138264(n) tends to C. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 09 2008
O.g.f.: 2x/3+7/9+(59x-7)/[9(1-8x+3x^2)]. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 24 2008
a(n) = 31*sqrt(13)*(A^n-B^n)/234-7*(A^n+B^n)/18 for n>1 where A=3/(4-sqrt(13)) and B=3/(4+sqrt(13)). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 24 2008
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EXAMPLE
| Starting with the configuration with bends (-1,2,2,3) with sum(bends) = 6, the next generation contains four circles with bends 3,6,6,15. The sum is 30 = 6*a(2). The third generation has 12 circles with sum(bends) = 234 = 6*a(3).
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MATHEMATICA
| CoefficientList[Series[(2 z^2 - 3 z + 1)/(3 z^2 - 8 z + 1), {z, 0, 100}], z] (* and *) LinearRecurrence[{8, -3}, {1, 5, 39}, 100] (* From Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)
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PROG
| (PARI) Vec((2*x^3 - 3*x^2 + x)/(3*x^2 - 8*x + 1)+O(x^99)) \\ Charles R Greathouse IV, Jul 03, 2011
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CROSSREFS
| Cf. A105970, A137246, A138264, A189226, A189227.
Sequence in context: A053573 A003482 A201442 * A105426 A115187 A129763
Adjacent sequences: A135846 A135847 A135848 * A135850 A135851 A135852
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KEYWORD
| easy,nonn
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AUTHOR
| Colin Mallows (colinm(AT)research.avayalabs.com), Mar 06 2008
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EXTENSIONS
| More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 24 2008
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