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Search: id:A137246
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| A137246 |
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a(n) is the ratio of the sum of the squares of the bends (curvatures) of the n-th generation of an Apollonian packing to the sum of the squares of the bends of the initial four-circle configuration. |
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+0
6
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| 1, 17, 339, 6729, 133563, 2651073, 52620771, 1044462201, 20731381707, 411494247537, 8167690805619, 162119333369769, 3217883594978523, 63871313899461153, 1267772627204287491, 25163838602387366361
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OFFSET
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1,2
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COMMENT
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These ratios are independent of the starting configuration. Similar ratios of third and higher moments are not so independent.
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REFERENCES
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J. C. Lagarias, C. L. Mallows and A. R. Wilks, American Mathematical Monthly (2002) 338-361.
J. C. Lagarias, C. L. Mallows and A. R. Wilks, Beyond the Descartes Circle Theorem, Amer. Math. Monthly, 109 (2002), 338-361.
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FORMULA
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For n >= 4, a(n) = 20a(n-1) - 3a(n-2)
O.g.f.: x*(2*x-1)*(x-1)/(1-20*x+3*x^2). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 31 2008
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EXAMPLE
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Starting with the configuration with bends (-1,2,2,3) with sum(bends^2) = 18, the next generation contains four circles with bends 3,6,6,15. The sum of their squares is 306 = 18*a(2). The third generation has 12 circles with sum(bends^2) = 6102 = 18*a(3).
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CROSSREFS
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Cf. A135849, A105970.
Adjacent sequences: A137243 A137244 A137245 this_sequence A137247 A137248 A137249
Sequence in context: A136270 A009046 A012112 this_sequence A081421 A121824 A120287
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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), Mar 09 2008
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EXTENSIONS
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More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Mar 31 2008
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