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A154637
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a(n) is the ratio of the sum of squares of the bends of the circles that are added in the n-th generation of Apollonian packing, to the sum of squares of the bends of the initial three circles.
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3
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1, 2, 66, 1314, 26082, 517698, 10275714, 203961186, 4048396578, 80356048002, 1594975770306, 31658447262114, 628384017931362, 12472705016840898, 247568948283023874, 4913960850609954786, 97536510167350024098, 1935988320795170617602, 38427156885401362279746, 762735172745641733742114
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OFFSET
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0,2
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COMMENTS
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For more references and links, see A189226.
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REFERENCES
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Colin Mallows, Growing Apollonian packings, J. Integer Sequences v.12, article 09.2.1 (2009).
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LINKS
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Table of n, a(n) for n=0..19.
Index to sequences with linear recurrences with constant coefficients, signature (20,-3).
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FORMULA
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G.f.: (1-18*x+29*x^2)/(1-20*x+3*x^2)
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EXAMPLE
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Starting with three circles with bends -1,2,2, the ssq is 9. The first derived generation has two circles, each with bend 3. So a(1) = (9+9)/9 = 2.
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MATHEMATICA
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CoefficientList[Series[(29 z^2 - 18 z + 1)/(3 z^2 - 20 z + 1), {z, 0, 100}], z] (* and *) LinearRecurrence[{20, -3}, {1, 2, 66}, 100] (* From Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)
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CROSSREFS
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For starting with four circles, see A137246. For sums of bends, see A135849 and A154636. For three dimensions, see A154638 - A154645.
Cf. also A189226, A189227.
Sequence in context: A098532 A159716 A157060 * A069865 A218433 A092884
Adjacent sequences: A154634 A154635 A154636 * A154638 A154639 A154640
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KEYWORD
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easy,nonn
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AUTHOR
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Colin Mallows (colinm(AT)research.avayalabs.com), Jan 13 2009
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EXTENSIONS
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More terms from N. J. A. Sloane, Nov 22 2009
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STATUS
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approved
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