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A105817
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Decimal expansion of the Fibonacci nested radical.
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9
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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
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OFFSET
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1,2
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COMMENTS
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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.
Clawson misstates Vijayaraghavan's theorem. Vijayaraghavan proved that for a_n > 0, the infinite radical sqrt(a_1 + sqrt(a_2 + sqrt(a_3 + ...))) converges if and only if limsup (log a_n)/2^n < infinity. (For example, suppose a_n = 1 if n is odd, and a_n = e^2^n if n is even. Then (log a_n)/2^n = 0, 1, 0, 1, 0, 1, ... for n >= 1, so the limit does not exist. However, limsup (log a_n)/2^n = 1 and the infinite radical converges.) - Jonathan Sondow, Mar 25 2014
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REFERENCES
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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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Jonathan M. Borwein and G. de Barra, Nested Radicals, Amer. Math. Monthly 98, 735-739, 1991.
Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.
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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]] (* Robert G. Wilson v, 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, A239349 for other nested radicals.
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KEYWORD
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AUTHOR
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STATUS
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approved
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