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Search: id:A135849
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| A135849 |
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a(n) is the ratio of the sum of the bends (curvatures) of the circles in the n-th generation of an Apollonian packing to the sum of the bends in the initial four-circle configuration. |
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5
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| 1, 5, 39, 297, 2259, 17181, 130671, 993825, 7558587, 57487221, 437222007, 3325314393, 25290849123, 192350849805, 1462934251071, 11126421459153, 84622568920011, 643601286982629, 4894942589100999, 37228736851860105
(list; graph; listen)
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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.
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REFERENCES
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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)=8a(n-1)-3a(n-2)
For n>2, [a(n+2), a(n+3)] = the 2 X 2 matrix [0,1; -3,8] * [5,39]. Example: [0,1; -3,8]^3 * [5,39] = [a(5), a(6)] = [2259, 17181]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 09 2008
a(n) = floor(C * A138264(n)), where C = 1.057097576... = (1/2)*((1/9) + sqrt((1/81) + 4)). Example: a(7) = 130671 = floor(C * A138264(7)) = floor(C * 123613). A135849(n)/A138264(n) tends to C. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 09 2008
O.g.f.: 2x/3+7/9+(59x-7)/[9(1-8x+3x^2)]. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 24 2008
a(n) = 31*sqrt(13)*(A^n-B^n)/234-7*(A^n+B^n)/18 for n>1 where A=3/(4-sqrt(13)) and B=3/(4+sqrt(13)). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 24 2008
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EXAMPLE
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Starting with the configuration with bends (-1,2,2,3) with sum(bends) = 6, the next generation contains four circles with bends 3,6,6,15. The sum is 30 = 6*a(2). The third generation has 12 circles with sum(bends) = 234 = 6*a(3).
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CROSSREFS
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Cf. A105970, A137246.
Cf. A138264.
Sequence in context: A075135 A053573 A003482 this_sequence A105426 A115187 A129763
Adjacent sequences: A135846 A135847 A135848 this_sequence A135850 A135851 A135852
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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 06 2008
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EXTENSIONS
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More terms from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 24 2008
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