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Search: id:A060790
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| A060790 |
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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. |
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+0
3
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| -1, 2, 2, 3, 15, 38, 110, 323, 927, 2682, 7754, 22403, 64751, 187134, 540822, 1563011, 4517183, 13054898, 37729362, 109039875, 315131087, 910745750, 2632104062, 7606921923, 21984412383, 63536130986, 183622826522, 530679817859
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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.
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REFERENCES
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Jeffrey C. Lagarias, Colin L. Mallows and Allan R. Wilks, Beyond the Descartes Circle Theorem, Jan 09 2001.
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LINKS
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Harry J. Smith, Table of n, a(n) for n=0,...,200
I. Peterson, Circle Game, Science News, 4/21/01.
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FORMULA
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a(n)=2a(n-1)+2a(n-2)+2a(n-3)-a(n-4).
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EXAMPLE
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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.
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PROGRAM
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(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]
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CROSSREFS
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Cf. A042944.
Sequence in context: A094352 A073828 A153938 this_sequence A109843 A164022 A089751
Adjacent sequences: A060787 A060788 A060789 this_sequence A060791 A060792 A060793
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KEYWORD
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easy,sign
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AUTHOR
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Brian L. Galebach (sequence(AT)ProbabilitySports.com), Apr 26 2001
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EXTENSIONS
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Corrected by T. D. Noe (noe(AT)sspectra.com), Nov 08 2006
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