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A135849 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. 8
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; text; internal format)
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), 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 (8,-3).

FORMULA

For n >= 4, a(n) = 8*a(n-1) - 3*a(n-2).

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*x-7)/(9*(1-8*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/(4-sqrt(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(n-1) - 3*Self(n-2): 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

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Last modified December 7 15:01 EST 2022. Contains 358667 sequences. (Running on oeis4.)