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 - (4-sqrt(13))^n*(7+sqrt(13)))) / (3*sqrt(13)) for n>0.
a(n) = 8*a(n-1) - 3*a(n-2) 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
KEYWORD
easy,nonn
AUTHOR
Colin Mallows, Jan 13 2009
EXTENSIONS
More terms from N. J. A. Sloane, Nov 22 2009
STATUS
approved