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A107905
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Decimal expansion of (5+sqrt(21))/2.
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1
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4, 7, 9, 1, 2, 8, 7, 8, 4, 7, 4, 7, 7, 9, 2, 0, 0, 0, 3, 2, 9, 4, 0, 2, 3, 5, 9, 6, 8, 6, 4, 0, 0, 4, 2, 4, 4, 4, 9, 2, 2, 2, 8, 2, 8, 8, 3, 8, 3, 9, 8, 5, 9, 5, 1, 3, 0, 3, 6, 2, 1, 0, 6, 1, 9, 5, 3, 4, 3, 4, 2, 1, 2, 7, 7, 3, 8, 8, 5, 4, 4, 3, 3, 0, 2, 1, 8, 0, 7, 7, 9, 7, 4, 6, 7, 2, 2, 5, 1, 6
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| From the Jin-Reidys paper: "In this paper we present the asymptotic enumeration of RNA structures with pseudoknots. We develop a general framework for the computation of exponential growth rate and the sub exponential factors for k-noncrossing RNA structures. Our results are based on the generating function for the number of k-noncrossing RNA pseudoknot structures, S_k(n)... where k-1 denotes the maximal size of sets of mutually intersecting bonds. We prove a functional equation for the generating function sum[n>= 0]S_k(n)z^n and obtain for k=2 and k=3 the analytic continuation and singular expansions, respectively. It is implicit in our results that for arbitrary k singular expansions exist and via transfer theorems of analytic combinatorics we obtain asymptotic expression for the coefficients. We explicitly derive the asymptotic expressions for 2- and 3-noncrossing RNA structures. Our main result is the derivation of the formula S_3(n) ~ ((10.4724 * 4!/(n(n-1)...(n-4))) * ((5+sqrt(21))/2})^n."
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REFERENCES
| D. Mumford et al., Indra's Pearls, Cambridge 2002; see p. 317. [From N. J. A. Sloane, Nov 22 2009]
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LINKS
| Emma Y. Jin and Christian M. Reidys, Asymptotic Enumeration of RNA Structures with Pseudoknots, Theorem 5, p. 15.
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EXAMPLE
| 4.7912878474779200032940235968640042444922282883839859513036...
The zeros at 15, 16 and 17 digits after the decimal point allow for a good rational approximation. The continued fraction is [4,1,3,1,3,1,3, ...] = 4 + 1/(1+ 1/(3+ 1/(1+ 1/(3+ 1/(1+ 1/(3+ 1(/1+ ...
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MATHEMATICA
| RealDigits[(5+Sqrt[21])/2, 10, 120][[1]] (* From Harvey P. Dale, May 02 2011 *)
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CROSSREFS
| Equals 1+A090458. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 24 2008]
Sequence in context: A190967 A004787 A191762 * A010299 A201413 A021680
Adjacent sequences: A107902 A107903 A107904 * A107906 A107907 A107908
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KEYWORD
| cons,easy,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Jun 22 2007
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