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Search: id:A105817
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| A105817 |
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Decimal expansion of the Fibonacci nested radical. |
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+0
3
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| 1, 6, 6, 1, 9, 8, 2, 4, 6, 2, 3, 2, 7, 8, 1, 1, 5, 5, 7, 9, 6, 7, 6, 0, 6, 0, 8, 1, 8, 1, 5, 1, 3, 1, 2, 9, 5, 0, 5, 6, 1, 6, 7, 5, 6, 2, 4, 6, 5, 0, 3, 5, 0, 0, 8, 2, 9, 9, 0, 6, 8, 0, 6, 7, 4, 3, 0, 6, 2, 9, 7, 2, 3, 5, 9, 8, 9, 5, 7, 3, 8, 1, 0, 8, 1, 7, 1, 6, 7, 0, 4, 1, 1, 0, 8, 4, 9, 2, 6, 6, 6, 9, 2, 2, 5
(list; cons; graph; listen)
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OFFSET
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1,2
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COMMENT
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The continued fraction expression of this is A105818. "It was discovered by T. Vijayaraghavan that the infinite radical, sqrt( a_1 + sqrt( a_2 + sqrt ( a_3 + sqrt( a_4 + ... where a_n => 0, will converge to a limit if and only if the limit of (ln a_n)/2^n exists." [Clawson, 229; Sloane]. We know the asymptotic limit of Fibonacci numbers is Phi^n (Binet expansion) and that Phi^n < 2^n and hence that the Fibonacci Nested Radical converges.
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REFERENCES
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Borwein, J. M. and de Barra, G. "Nested Radicals." Amer. Math. Monthly 98, 735-739, 1991.
Calvin C. Clawson, "Mathematical Mysteries, the beauty and magic of numbers," Perseus Books, Cambridge, Mass., 1996, pages 142 & 229.
S. R. Finch, "Analysis of a Radical Expansion." Section 1.2.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, p. 8, 2003.
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LINKS
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Eric Weisstein's World of Mathematics, Nested Radical Constant.
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FORMULA
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Sqrt(1 + Sqrt(1 + Sqrt(2 + Sqrt(3 + Sqrt(5 + ... + Sqrt(Fibonacci(n)=A000045)))).
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EXAMPLE
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1.66198246232781155796760608181513129505616756246503500829906806743...
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MATHEMATICA
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RealDigits[ Fold[ Sqrt[ #1 + #2] &, 0, Reverse[ Fibonacci[ Range[50]]]], 10, 111][[1]] (from Robert G. Wilson v (rgwv(AT)rgwv.com), Apr 21 2005)
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CROSSREFS
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Cf. A000045; A072449, A083869, A099874, A099876, A099877, A099878, A099879, A105546, A105548, A105815, A105816, A105818 for other nested radicals.
Cf. A151558.
Sequence in context: A112302 A073012 A102522 this_sequence A093313 A098267 A122193
Adjacent sequences: A105814 A105815 A105816 this_sequence A105818 A105819 A105820
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KEYWORD
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cons,nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Apr 21 2005
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