

A135849


a(n) is the ratio of the sum of the bends (curvatures) of the circles in the nth generation of an Apollonian packing to the sum of the bends in the initial fourcircle configuration.


8



1, 5, 39, 297, 2259, 17181, 130671, 993825, 7558587, 57487221, 437222007, 3325314393, 25290849123, 192350849805, 1462934251071, 11126421459153, 84622568920011, 643601286982629, 4894942589100999, 37228736851860105, 283145067047577843, 2153474325825042429
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OFFSET

1,2


COMMENTS

These ratios are independent of the starting configuration.
For more comments, references and links, see A189226.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200
J. C. Lagarias, C. L. Mallows and Allan Wilks, Beyond the Descartes Circle Theorem, Amer. Math. Monthly, 109 (2002), 338361.
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 (8,3).


FORMULA

For n >= 4, a(n) = 8*a(n1)  3*a(n2).
For n>2, [a(n+2), a(n+3)] = the 2 X 2 matrix [0,1; 3,8]^n * [5,39]. Example: [0,1; 3,8]^3 * [5,39] = [a(5), a(6)] = [2259, 17181].  Gary W. Adamson, Mar 09 2008 (typo corrected by Jonathan Sondow, Dec 24 2012)
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, Mar 09 2008
O.g.f.: 2*x/3 +7/9 +(59*x7)/(9*(18*x+3*x^2)).  R. J. Mathar, 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/(4sqrt(13)) and B=3/(4+sqrt(13)).  R. J. Mathar, Apr 24 2008


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).


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] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)


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
(Magma) I:=[1, 5, 39]; [n le 3 select I[n] else 8*Self(n1)  3*Self(n2): n in [1..30]]; // Vincenzo Librandi, Dec 25 2012


CROSSREFS

Cf. A105970, A137246, A138264, A189226, A189227.
Sequence in context: A003482 A221357 A201442 * A105426 A356622 A273019
Adjacent sequences: A135846 A135847 A135848 * A135850 A135851 A135852


KEYWORD

easy,nonn


AUTHOR

Colin Mallows, Mar 06 2008


STATUS

approved



