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Search: id:A105818
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| A105818 |
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Continued fraction expansion of the Fibonacci nested radical (A105817). |
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+0
2
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| 1, 1, 1, 1, 23, 18, 1, 1, 1, 1, 1, 1, 2, 1, 22, 2, 1, 53, 1, 1, 10, 1, 1, 17, 2, 4, 1, 27, 1, 2, 422, 3, 3, 13, 12, 5, 28, 1, 3, 1, 2, 1, 3, 2, 4, 6, 6, 3, 5, 50, 1, 1, 6, 3, 2, 1, 118, 2, 1, 1, 2, 6, 1, 4, 1, 1, 5, 2, 3, 3, 16, 1, 4, 6, 2, 2, 22, 4, 3, 10, 1, 1, 49, 5, 1, 1, 12, 1, 1, 3, 13, 3, 10, 1, 2
(list; graph; listen)
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OFFSET
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1,5
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COMMENT
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The decimal expression of this is A105817. "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.
Finch, S. R. "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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f[n_] := Block[{k = n, s = 0}, While[k > 0, s = Sqrt[s + Fibonacci[k]]; k-- ]; s]; ContinuedFraction[ f[46], 95] (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, A105817 for other nested radicals.
Sequence in context: A124601 A040508 A070716 this_sequence A085450 A077146 A077576
Adjacent sequences: A105815 A105816 A105817 this_sequence A105819 A105820 A105821
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KEYWORD
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cofr,nonn
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AUTHOR
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Jonathan Vos Post (jvospost2(AT)yahoo.com), Apr 21 2005
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