OFFSET
0,2
COMMENTS
The ratio of successive terms approaches the constant phi+sqrt(phi) ~= 2.89005363826396..., where phi is the golden ratio (sqrt(5)+1)/2. The ratio between the curvatures of two successively smaller circles approaches this constant in any apollonian packing as the curvatures increase.
For more comments, references and links, see A189226.
LINKS
Harry J. Smith, Table of n, a(n) for n = 0..200
Jeffrey C. Lagarias, Colin L. Mallows and Allan Wilks, Beyond the Descartes Circle Theorem, arXiv:math/0101066 [math.MG], 2001.
Jeffrey C. Lagarias, Colin L. Mallows and Allan Wilks, Beyond the Descartes Circle Theorem, Amer. Math Monthly, 109 (2002), 338-361.
I. Peterson, Circle Game, Science News, 4/21/01.
Index entries for linear recurrences with constant coefficients, signature (2, 2, 2, -1).
FORMULA
a(n) = 2a(n-1) + 2a(n-2) + 2a(n-3) - a(n-4).
G.f.: -(1-x)*(1 - 3*x - 3*x^2)/(1 - 2*x - 2*x^2 - 2*x^3 + x^4). - Colin Barker, Apr 22 2012
Lim_{n -> inf} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018
EXAMPLE
After circles of 2, 2, 3, 15 have been inscribed in the diagram, the next smallest circle that can be inscribed has a curvature of 38.
MAPLE
seq(coeff(series((x-1)*(1-3*x-3*x^2)/(1-2*x-2*x^2-2*x^3+x^4), x, n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Sep 12 2018
MATHEMATICA
CoefficientList[Series[(-3 z^3 + 4 z - 1)/(z^4 - 2 z^3 - 2 z^2 - 2 z + 1), {z, 0, 100}], z] (* and *) LinearRecurrence[{2, 2, 2, -1}, {-1, 2, 2, 3}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)
PROG
(PARI) { for (n=0, 200, if (n>3, a=2*a1 + 2*a2 + 2*a3 - a4; a4=a3; a3=a2; a2=a1; a1=a, if (n==0, a=a4=-1, if (n==1, a=a3=2, if (n==2, a=a2=2, a=a1=3)))); write("b060790.txt", n, " ", a); ) } \\ Harry J. Smith, Jul 12 2009
(GAP) a:=[-1, 2, 2, 3];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]+2*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Sep 12 2018
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Brian Galebach, Apr 26 2001
EXTENSIONS
Corrected by T. D. Noe, Nov 08 2006
STATUS
approved