Search: id:A060790
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%I A060790
%S A060790 1,2,2,3,15,38,110,323,927,2682,7754,22403,64751,187134,540822,1563011,
%T A060790 4517183,13054898,37729362,109039875,315131087,910745750,2632104062,
%U A060790 7606921923,21984412383,63536130986,183622826522,530679817859
%V A060790 -1,2,2,3,15,38,110,323,927,2682,7754,22403,64751,187134,540822,1563011,
               4517183,
%W A060790 13054898,37729362,109039875,315131087,910745750,2632104062,7606921923,
               21984412383,
%X A060790 63536130986,183622826522,530679817859
%N A060790 Inscribe two circles of curvature 2 inside a circle of curvature -1. 
               Sequence gives curvatures of the smallest circles that can be sequentially 
               inscribed in such a diagram.
%C A060790 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.
%D A060790 Jeffrey C. Lagarias, Colin L. Mallows and Allan R. Wilks, Beyond the 
               Descartes Circle Theorem, Jan 09 2001.
%H A060790 Harry J. Smith, <a href="b060790.txt">Table of n, a(n) for n=0,...,200</
               a>
%H A060790 I. Peterson, <a href="http://www.sciencenews.org/20010421/bob18.asp">
               Circle Game</a>, Science News, 4/21/01.
%F A060790 a(n)=2a(n-1)+2a(n-2)+2a(n-3)-a(n-4).
%e A060790 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.
%o A060790 (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); ) } [From Harry J. Smith 
               (hjsmithh(AT)sbcglobal.net), Jul 12 2009]
%Y A060790 Cf. A042944.
%Y A060790 Sequence in context: A094352 A073828 A153938 this_sequence A109843 A164022 
               A089751
%Y A060790 Adjacent sequences: A060787 A060788 A060789 this_sequence A060791 A060792 
               A060793
%K A060790 easy,sign
%O A060790 0,2
%A A060790 Brian L. Galebach (sequence(AT)ProbabilitySports.com), Apr 26 2001
%E A060790 Corrected by T. D. Noe (noe(AT)sspectra.com), Nov 08 2006

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