

A154636


a(n) is the ratio of the sum of the bends of the circles that are drawn in the nth generation of Apollonian packing to the sum of the bends of the circles in the initial configuration of 3 circles.


6



1, 2, 18, 138, 1050, 7986, 60738, 461946, 3513354, 26720994, 203227890, 1545660138, 11755597434, 89407799058, 679995600162, 5171741404122, 39333944432490, 299156331247554, 2275248816682962, 17304521539721034, 131610425867719386, 1000969842322591986
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OFFSET

0,2


COMMENTS

For comments and more references and links, see A189226.


LINKS

Colin Barker, Table of n, a(n) for n = 0..1000
C. L. Mallows, Growing Apollonian Packings, J. Integer Sequences, 12 (2009), article 09.2.1.
Index entries for linear recurrences with constant coefficients, signature (8,3).


FORMULA

G.f.: (1  x)*(1  5*x) / (1  8*x + 3*x^2).
From Colin Barker, Jul 15 2017: (Start)
a(n) = (((7+sqrt(13))*(4+sqrt(13))^n  (4sqrt(13))^n*(7+sqrt(13)))) / (3*sqrt(13)) for n>0.
a(n) = 8*a(n1)  3*a(n2) for n>2.
(End)


EXAMPLE

Starting from three circles with bends 1,2,2 summing to 3, the first derived generation consists of two circles, each with bend 3. So a(1) is (3+3)/3 = 2.


MATHEMATICA

CoefficientList[Series[(5 z^2  6 z + 1)/(3 z^2  8 z + 1), {z, 0, 100}], z] (* and *) LinearRecurrence[{8, 3}, {1, 2, 18}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)


PROG

(PARI) Vec((1  x)*(1  5*x) / (1  8*x + 3*x^2) + O(x^30)) \\ Colin Barker, Jul 15 2017


CROSSREFS

Other sequences relating to the twodimensional case are A135849, A137246, A154637. For the threedim. case see A154638  A154645. Five dimensions: A154635.
Cf. also A189226, A189227.
Sequence in context: A057971 A073512 A005544 * A216584 A193446 A226733
Adjacent sequences: A154633 A154634 A154635 * A154637 A154638 A154639


KEYWORD

easy,nonn


AUTHOR

Colin Mallows, Jan 13 2009


EXTENSIONS

More terms from N. J. A. Sloane, Nov 22 2009


STATUS

approved



